Wikiversity:Colloquium

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

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

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

== Proposal to rehost Wikinews here ==

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

I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:30, 14 May 2026 (UTC)
:I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:14, 9 June 2026 (UTC)

===Votes===
*{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 05:35, 14 May 2026 (UTC)
*{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC)
*{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of &quot;practicing journalism&quot; a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC)
*{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 15 May 2026 (UTC)
*{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC)
*{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 20:29, 30 May 2026 (UTC)
* {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC)
*{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC)
* {{oppose}} per above. Wikiversity&lt;math&gt;\not=&lt;/math&gt; Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC)
* {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC)
* {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC)
*{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on &quot;Media &amp; Democracy&quot; syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that &quot;rehost Wikinews here&quot; would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}}

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

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

:Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 15 May 2026 (UTC)
*Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;''' Blue Rasberry '''&lt;/span&gt;]][[User talk:Bluerasberry|&lt;span style=&quot;cursor:help&quot;&gt;&lt;span style=&quot;background:#cedff2;color:#11e&quot;&gt;(talk)&lt;/span&gt;&lt;/span&gt;]] 19:17, 22 May 2026 (UTC)
*:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 21:15, 22 May 2026 (UTC)
*::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a &quot;regular&quot; journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: #0000FF;&quot;&gt;OhanaUnited&lt;/span&gt;&lt;/b&gt;]][[User talk:OhanaUnited|&lt;b&gt;&lt;span style=&quot;color: green;&quot;&gt;&lt;sup&gt;Talk page&lt;/sup&gt;&lt;/span&gt;&lt;/b&gt;]] 22:43, 5 June 2026 (UTC)
*:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC)

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

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

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

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

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

== May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan ==
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Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions.

#'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe
#'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific

Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally,  Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. 

Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]!
&lt;br /&gt;
[[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]&lt;/div&gt;
&lt;span style=&quot;white-space:nowrap;&quot;&gt;[[User:Pine|&lt;span style=&quot;color:#01796f; text-shadow:#00BFFF 0 0 1.0em&quot;&gt;↠Pine&lt;/span&gt;]] [[User talk:Pine|&lt;span style=&quot;color:DeepSkyBlue&quot;&gt;(&lt;b style=&quot;color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em&quot;&gt;✉&lt;/b&gt;)&lt;/span&gt;]]&lt;/span&gt; 19:46, 21 May 2026 (UTC)

== Vote now in the 2026 U4C election ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC].

Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,&lt;section end=&quot;announcement-content&quot; /&gt;

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

== Create an autopatrolled user group? ==

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

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

:'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 15:45, 29 May 2026 (UTC)
:: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC)
:::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:42, 29 May 2026 (UTC)
:::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC)
:::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:16, 30 May 2026 (UTC)
: '''Support''' - sounds like a good idea
:* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: &quot;Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors.&quot;
:* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects?
: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:28, 30 May 2026 (UTC)
::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]].
::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only.
:: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC)
: @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC)
::Thanks. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:13, 9 June 2026 (UTC)

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

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

:We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 08:50, 4 June 2026 (UTC)
:Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC)
::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC)
:::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC)
::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC)

== Draft inactivity policy ==

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

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

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

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

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

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

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC)

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

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

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

== Towards an Ethics policy ==

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

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

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

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

Hello community,

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

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

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

:Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:24, 9 June 2026 (UTC)

== Image not displaying ==

Can anyone work out why this image isn't displaying?&lt;br&gt;
[[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:45, 11 June 2026 (UTC)</text>
      <sha1>rk9i2fhwi9x62q5qxs1ix3kq1tz0jhr</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Userboxes</title>
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      <contributor>
        <username>AUBSTRAWBS</username>
        <id>3060598</id>
      </contributor>
      <comment>added link for anime user boxes</comment>
      <origin>2815122</origin>
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      <text bytes="3131" sha1="nqy097r71xdu74cele92n9bj38jj899" xml:space="preserve">{{shortcut|WV:UBX|WV:USERBOX}}
{{Administering Wikiversity}}
In order to promote collaborative learning within the Wikiversity community it is relevant to know the interests of other users. '''The userboxes''' can be a useful way to introduce yourself to the community. Please make responsible use of these templates to develop the Wikiversity community. Templates that are counter to the educational mission of Wikiversity or otherwise disrupt the community will be deleted.

==Users by language skills==
See the sub-categories of [[:Category:Users by language]] for a complete list of language userboxes.

{|
!Adding this to your page!!Creates this
|-
|&lt;nowiki&gt;{{User en}}&lt;/nowiki&gt;||{{User en}}
|-
|&lt;nowiki&gt;{{User de}}&lt;/nowiki&gt;||{{User de}}
|-
|&lt;nowiki&gt;{{User fr}}&lt;/nowiki&gt;||{{User fr}}
|-
|&lt;nowiki&gt;{{User es}}&lt;/nowiki&gt;||{{User es}}
|}

== Involvement of learners ==

{|
!Adding this to your page!!Creates this
|-
|&lt;nowiki&gt;{{NLP}}&lt;/nowiki&gt;||{{NLP}}
|-
|&lt;nowiki&gt;{{User Chemcontrib}}&lt;/nowiki&gt;||{{User Chemcontrib}}
|-
|&lt;nowiki&gt;{{User Mathcontrib}}&lt;/nowiki&gt;||{{User Mathcontrib}}
|-
|&lt;nowiki&gt;{{User Strategycontrib}}&lt;/nowiki&gt;||{{User Strategycontrib}}
|-
|&lt;nowiki&gt;{{User Gamescontrib}}&lt;/nowiki&gt;||{{User Gamescontrib}}
|-
|&lt;nowiki&gt;{{Template:User DORA}}&lt;/nowiki&gt;||{{Template:User DORA}}
|-
|&lt;nowiki&gt;{{Wikipedia}}&lt;/nowiki&gt;||{{Wikipedia}}
|-
|&lt;nowiki&gt;{{User ITcontrib}}&lt;/nowiki&gt;||{{User ITcontrib}}
|}

==[[:Category:Knowledge user templates|Knowledge user templates]]==

Knowledge userboxes (or user templates) are used to advertise your level of knowledge in a certain subject.  You can also use them to find other students of that subject.
{|
!Adding this to your page!!Creates this
|-
|&lt;nowiki&gt;{{User knowledge-stst-0}}&lt;/nowiki&gt;||{{User knowledge-stst-0}}
|-
|&lt;nowiki&gt;{{User knowledge-agri-1}}&lt;/nowiki&gt;||{{User knowledge-agri-1}}
|-
|&lt;nowiki&gt;{{User knowledge-pols-2}}&lt;/nowiki&gt;||{{User knowledge-pols-2}}
|-
|&lt;nowiki&gt;{{User knowledge-educ-3}}&lt;/nowiki&gt;||{{User knowledge-educ-3}}
|-
|&lt;nowiki&gt;{{User knowledge-lang-4}}&lt;/nowiki&gt;||{{User knowledge-lang-4}}
|-
|}

'''More''': [[:Category:Knowledge user templates]].

== Users by Wikimedia project participation ==

{|
!Adding this to your page!!Creates this
|-
|&lt;nowiki&gt;{{User Commons}}&lt;/nowiki&gt;||{{User Commons}}
|-
|&lt;nowiki&gt;{{User Wikipedia}}&lt;/nowiki&gt;||{{User Wikipedia}}
|-
|&lt;nowiki&gt;{{User Wikiversity Beta}}&lt;/nowiki&gt;||{{User Wikiversity Beta}}
|-
|}


== User characteristics ==

{|
!Adding this to your page!!Creates this
|-
|&lt;nowiki&gt;{{User must have gone crazy|[[link|text]]}}&lt;/nowiki&gt;&lt;br /&gt;&lt;br /&gt;link: can be a page inside Wikiversity&lt;br /&gt;text: is a description for link||{{User must have gone crazy}}
|-
|&lt;nowiki&gt;{{Google video chat user}}&lt;/nowiki&gt;&lt;br /&gt;&lt;br /&gt;This user can be reached by [[w:Google_Voice|Google video chat]]||{{Google video chat user}}
|-
|&lt;nowiki&gt;{{Logitech Vid user}}&lt;/nowiki&gt;&lt;br /&gt;&lt;br /&gt;This user can be reached by [[w:SightSpeed|Logitech Vid]]||{{Logitech Vid user}}
|-
|}

== Anime and Manga ==
To find '''[[anime]]''' and manga userboxes go to '''[[Wikiversity:Userboxes/anime]]'''

==See also==
{{Wikiversity culture}}

[[Category:Wikiversity culture]]</text>
      <sha1>nqy097r71xdu74cele92n9bj38jj899</sha1>
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  </page>
  <page>
    <title>Meditation</title>
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      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>+[[Meditation: An Overview and Analysis]]</comment>
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      <text bytes="12368" sha1="3vboh5wwt53gzu6hl1o3iafs3mdc9s3" xml:space="preserve">[[w:Meditation|Meditation]] is a way of calming thoughts, relaxation, spiritual healing and enlightenment. 

==Meditation for Beginners==
[[File:Meditation for Beginners.ogv|thumb|Easy meditation for beginners (6 min.)]]
Sit down comfortably. The hands are in the lap or on the legs. The back is straight. The abdomen is relaxed.

1. Concentrate on your body. Think several times (5-10 times) the mantra &quot;Om&quot; in the head, chest, abdomen, hands, legs, feet, in the ground and then in the whole universe: &quot;Om Shanti. Om Peace ... &quot;

2. Stop one minute every thought. When thoughts come, push them off. 

3. Linger one minute in the meditation. Thoughts may come and go as they want.

4. Relax. Feel the peace and relaxation in your body.

5. Move your body, your feet and your hands. You're back. Ahead with optimism.

-&gt; Video: [http://www.youtube.com/watch?v=RbdKfTFvrEk Meditation for Beginners (Youtube, 6 min)]

== Computer Meditation ==
[[File:Computer Meditation.ogv|thumb|Computer Meditation (5 min.)]]
A guided meditation on the computer is a fast way of relaxation. Through simple yoga movements in connection with positive sentences you can resolve emotional blocks and get a positive spirit in five minutes. See also the [http://www.youtube.com/watch?v=XJrp2W77sIg video.]

1. Visualize yourself as a Master of life (Buddha, Shiva, Goddess) and rub the earth with your feet, &quot;I am a Master of life. I go my way positively. &quot;

2. Move a hand and think: &quot;I send light to ... (name). May all people be happy. May the world be happy.&quot;

3. Connect with the enlightenment energy (God, enlightened Masters), rub your palm in front of your heart and think, &quot;Om all enlightened Masters. Om inner wisdom. Please guide and help me on my way.&quot;

4. Get inner peace, rub the palms above your head and think, &quot;I take things as they are. I let my false desires go off.&quot; 

5.  Put your hands on your legs. The spine is straight and the belly is relaxed. Stop one minute all your thinking. Sit only there. Don´t think. Then relax. 

== Yoga-Meditation ==
[[File:Yoga-Meditation Yogi Nils.ogv|thumb|Yoga-Meditation (8 min.)]][[File:Yoga-Meditation.jpg|thumb|150px]]
We sit upright in our meditation seat (cross-legged, heel seat, chair) and in our meditation posture (hands on the legs or lap). The back is straight and the belly is relaxed.  The eyes are open, semi-open or closed.

1. Sun = We see a beautiful sun in the sky. She sends her rays down on us. We wrap our whole body with light and think the mantra &quot;Light.&quot;  We take a golden ray of sunshine and wrap ourselves with it completely. We circle with the light around our body and think the mantra &quot;Light, light, light.&quot; We let the light flow into us. We fill our entire body with light and think often the mantra &quot;Light.&quot;

2. Body = We bring our mind to rest by a mantra. We think the numbers 1 - 20 in the head, chest, abdomen, feet and into the earth.

3. Cosmos = We make big circles with our arms, visualize the whole universe full of stars around us and think the numbers 1 - 20 in the whole universe.

4. Blessing = We move a hand in blessing and think, &quot;I send light to (name). May all people be happy. May the world be happy.&quot;

5. Master = We rub the palms in front of the heart chakra, connecting us with the enlightened Masters and think, &quot;Om all enlightened Masters. Om inner wisdom. Please guide and help me on my way.&quot;

6. Mantra Om = We put your hands in our lap. We think a minute when breathing out (at breathing in, or at the inhalation and exhalation) the mantra &quot;Om&quot; in the belly.

7. Stop thinking = We stop a minute every thought. If thoughts come, we push them away. Then we relax for a minute.

Oracle = Forward with optimism. You are guided by the light (by the enlightened Masters, by God). Follow consistent your inner wisdom and your love. Make every day your spiritual exercises. Bring the light on your way into the world. Success.
* [[commons:File:Yoga-Meditation_Yogi_Nils.ogv|Video Yoga-Meditation]]

== Progressive Muscle Relaxation ==

[[w:Progressive muscle relaxation|Progressive muscle relaxation]] is a relaxation technique by alternately tensing and relaxing the muscles. The presented form was developed by Yogi Nils in his Yoga groups. We can do it in sitting or lying down.

1. Muscle tension = We tense the muscles of the legs and feet. We stop every thought. We can hold our breath or breathe normally. Then we relax the muscles of the legs and feet. We tense and relax in the same way the arms and hands, the head (face) and the whole body. 

2. Numbers = We count several times the numbers from 1 to 20 in the head, focusing on the head and breathe into the head. Our mind calms. We focus on the chest, breathe into the chest and count the numbers 1 to 20 in the chest. We breathe in the belly, and count there the numbers 1 to 20. We focus on the legs and feet, and count there the numbers 1 to 20. We visualize under the soles a large ball and count the numbers from 1 to 20 in the ball.

3. Sun = We see in the sky a beautiful sun. She sends her rays down on us. We feel her light and warmth on our skin. It is as if we are on holiday in the sun. We enjoy the sunlight. We wrap our whole body with light. We take a golden ray of sun and let the sunlight everywhere circle around us. We think the mantra &quot;Light.&quot; We let the sunlight flow into us and fill us with light. We think the mantra &quot;Light.&quot;

4. Sending light = We move a hand and send another person light. We envelop him with light and let the light flow into him. Think many times the word &quot;Light&quot;. After that we send light all over the world. We wrap the whole world with light, fill it with light and think often &quot;Light.&quot;

5. Stop thinking = We stop a minute every thought. Then we linger in meditation. Thoughts may come now if they want to. We feel our the body and enjoy the peace and well-being. If we have meditated enough, we come back again. We move our feet and hands. We stretch and move ourselves. We sit up and are back.

== Sun Meditation==
[[File:Sun-Meditation with Yogi Nils.ogv|thumb|Sun-Meditation (12 min.)]]
This is the main meditation in the yoga groups of Yogi Nils. It finishes each yoga class. We lay or sit down comfortably. Cover you for lying, so you do not get cold. 

1. Relaxation = We tense the muscles of the legs and feet. We keep the tension,  stop all thoughts and breathe into the legs. Then we relax. We stretch the muscles of the arms and hands. We breathe into the arms and hands. We relax. We tense the muscles of the head and face. We breathe into the face. We relax. We tense the muscles of the whole body. We breathe into the whole body. We relax.

2. Numbers  = We count several times the numbers from 1 to 20 in the head, focusing on the head and breathe into the head. Our mind calms. We focus on the chest, breathe into the chest and count the numbers 1 to 20 in the chest. We breathe in the belly, and count there the numbers 1 to 20. We focus on the legs and feet, and count there the numbers 1 to 20. We visualize under the soles a large ball and count the numbers from 1 to 20 in the ball.

3. Sun = We see in the sky a beautiful sun. She sends her rays down on us. We feel her light and warmth on our skin. It is as if we are on holiday in the sun. We enjoy the sunlight. We wrap our whole body with light. We take a golden ray of sun and let the sunlight everywhere circle around us. We think the mantra &quot;Light.&quot; We let the sunlight flow into us and fill us with light. We think the mantra &quot;Light.&quot;

4. Sending light = We move a hand and send another person light. We envelop him with light and let the light flow into him. Think many times the word &quot;Light&quot;. After that we send light all over the world. We wrap the whole world with light, fill it with light and think often &quot;Light.&quot;

5. Om Shanti = We think inhaling &quot;Om&quot; and exhaling &quot;Shanti&quot;. We stop all other thoughts. We feel calmness, serenity and peace in us. We stop a minute every thought and move gently our feet. We focus on our feet and move on, until our mind comes completely to rest. We relax. We lie a few minutes relaxed just there. We are in harmony with ourselves, our life and our world.

6. Coming back = We come back slowly. We move our feet and hands. We stretch and move ourselves. We sit up and are back. Life can come. We have the sun within us.

==Mantra Meditation (Easy Relaxation)==
[[File:Meditation and yoga in lying (easy relaxation).ogv|thumb|Mantra Meditation (12 min.)]]
[[File:Kundalini Meditation.ogv|thumb|Kundalini Meditation. Video 10 min.]]
We lie down comfortably. We can do this meditation in bed for a long time while listening to beautiful music. It is good for all persons who cannot slep or want to relax in bed. 

1. Knee to chest (arms around) with mantra: &quot;Om all enlightened Masters. Om inner wisdom. Please guide and help me on my way.&quot;

2. Prone position: &quot;I send light to (name). May all people be happy. May the world be happy.&quot; Hands under your head. Move your pelvis (spine) and move your feet.

3. Turn right in the Child Pose: &quot;Om Shanti&quot;. The hands lie on the legs or one hand on the legs (or the hip) and one hand under the head (stops your thoughts).

4. Turn left in the Child Pose: &quot;Om Shanti&quot;. The hands lie on the legs or one hand on the legs (or the hip) and one hand under the head (stops your thoughts).

5. Supine position, left feet on the right leg: &quot;Om Shanti&quot;. You can move the feet and the hands a little bit.

6. Right feet on the left leg. Move your feet and your hands: &quot;Om Shanti&quot;.  

7. Legs extended. Move the feet. Count the numbers 1 to 20 in the head, thorax, belly, legs and feet.

8. Move your hands in circling, visualize the universe and think: &quot;Om Universe&quot;. Relax.

9. Relax completely for some minutes. Hands at the sides or on the belly. 

10. Sit up and do the Buddy Breathing. Press with the thumb the right nostril and inhale through the left nostril. Press the left nostril and inhale at the right side. Make this several times until inner peace occurs. Then go back in your everyday life.
==Guided Mediations==
The Wikiversity resource on [[Guided Meditations]] provides links to instructions and scripts for several guided meditations.

==Wikimedia sound files==
&lt;!--Do not change this header without also updating sister link at [[Wikibooks:Yoga/Meditation#See_also]]--&gt;
[[File:Om pro.ogg|left]]&lt;nowiki&gt; &lt;/nowiki&gt;[[Wikipedia:Om]]
[[File:Om tat sat pro.ogg|left]]&lt;nowiki&gt; &lt;/nowiki&gt;[[Wikipedia:Om Tat Sat]]
[[File:Om Namo Bhagavate Vasudevaya Pronunciation.oga|left]]&lt;nowiki&gt; &lt;/nowiki&gt;[[Wikipedia:Om Namo Bhagavate Vasudevaya]]
[[File:Om Tare Tutare Ture Soha.vorb.oga|left]]&lt;nowiki&gt; &lt;/nowiki&gt;[[w:Special:Permalink/713230506#Some_other_mantras_in_Tibetan_Buddhism|Om Tare Tutare Ture Soha]] is a Tibetan Buddhist mantra
[[File:Ham'so mantra.vorb.oga|left]]&lt;nowiki&gt;  &lt;/nowiki&gt;[[c:File:Ham'so mantra.vorb.oga#Additional information|Ham'so]] is apparently both [[c:Category:Buddhist mantras|Buddhist]] and [[w:Soham (Sanskrit)|Hindu]].  The &quot;Ham&quot; sound is made on the inhalatione air, while &quot;so&quot; is done on the exhalation. The beginning and the end are not set, thus the mantra can be &quot;ham'so&quot; or &quot;so'ham&quot; without changing its meaning.&lt;ref&gt;[[Commons:Special:Permalink/67420700]]&lt;/ref&gt;

==Meditation Video==
[[File:Meditation with the inner voice.ogv|thumb|Meditation with the inner voice. Video 10 min.]]
*[http://www.youtube.com/watch?v=hbwTvZFIkIg Meditation with Yogi Nils (germ., 6 min.)] Relax with a guided meditation with beautiful music.
*[http://www.youtube.com/watch?v=F6eFFCi12v8 One-Moment Meditation: &quot;How to Meditate in a Moment&quot;]

==See also==
{{wikibooks|Yoga/Meditation}}
*[https://www.smashwords.com/books/view/245377 Download Wikibook Yoga with Yoga Oracle in various formats (PDF, Epub, Kindle).]
*[[Relaxation techniques]]
*[[b:God and Religious Toleration/Paradise Meditation|Paradise Meditation (Wikibooks)]]
*[http://itunes.apple.com/us/app/enso/id484567469?l=en&amp;ls=1&amp;mt=8 Easy start meditation on iPad]
* Wikipedia template [[w:Template:Meditation|''Meditation'']] (and to a lesser extent template [[w:Template:Hypnosis|''Hypnosis'']])
* [[Mindfulness]] and [[Self-regulation]] as related practices.
* [[Meditation: An Overview and Analysis]]

==References==
&lt;references/&gt;
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== Goals ==
The goal of the [[Educational Media Awareness Campaign]] is twofold:

# To help educators effectively use the internet to find suitable media for learning resources.
# To assist educators in integrating media correctly and legally, ensuring reusability and proper permission handling.

This campaign is primarily targeted at the primary and secondary education sectors, but it is also beneficial for tertiary, informal, and other educational fields.

== Background ==
One of the biggest challenges faced by creators of digital educational resources is the ''illegal use of images and graphic content''.

Publishers—both digital and print—regularly receive submissions that must be rejected due to copyright violations. Image copyright infringement is especially common in schools.

This situation is paradoxical. In recent years, the availability of well-documented, reusable, and redistributable media has increased significantly:

* Wikimedia Commons provides millions of legally reusable, well-categorized images.
* Flickr also offers a vast collection of reusable images, though less structured.

With such resources available, there is no need for illegal use of images.

The [[Educational Media Awareness Campaign]] aims to solve this issue by:

* Showcasing galleries of reusable images  
* Providing case studies on finding appropriate media  
* Listing trusted media repositories  
* Offering tutorials on licensing and documentation  

A key focus is introducing educators to the effective educational use of Wikimedia Commons.

== A Multi-site Effort ==
The [[Educational Media Awareness Campaign]] is a subproject of Wikiversity Outreach. It collaborates with various educational and media platforms to expand its impact.

== Galleries and Pictures of the Day ==
The campaign includes:

* 100+ featured images  
* Coverage across 12 major school subjects  
* Carefully written captions with source links  
* Access to thousands of related images  

These images serve as entry points for educators.

Additionally:

* Images are organized into dynamic pages  
* They can be used as &quot;Picture of the Day&quot; on wikis  
* Separate rotating sections exist for each subject  

== Investigation and Analysis of Digital Educational Resources ==
With technological advancement, digital educational resources have become increasingly diverse.

A survey of 82 undergraduate students revealed:

* Students prefer digital resources for self-directed learning  
* Search engines are the most commonly used access method  
* Traditional computers are still the preferred device  

To maximize effectiveness, educational providers should:

* Understand user behavior  
* Align resources with user needs  
* Optimize accessibility and usability

== See also ==
* [https://outreach.wikimedia.org/wiki/Education/Archive/Main_page Outreach:Education]

== References ==
&lt;references /&gt;

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'''{{big3|Table of contents}}'''
==Front matter==
# [[/Front cover/]] - [[User:Jtneill|Jtneill]]
# [[/Acknowledgements/]] - [[User:Jtneill|Jtneill]]
# [[/Preface/]] - [[User:Jtneill|Jtneill]]

==Introduction==
# [[/Introduction/What is motivation?|What is motivation?]] - [[User:Jtneill|Jtneill]]
# [[/Introduction/What is emotion?|What is emotion?]] - [[User:Jtneill|Jtneill]]
# [[/Introduction/Relationship between motivation and emotion|Relationship between motivation and emotion]] - [[User:Jtneill|Jtneill]]

==Motivation==
[[File:Goals affirmation poster, Navy · DF-SD-04-09850.JPEG|right|160px]]
[[File:Love Rush!.jpg|right|160px]]
# Biological motives (needs)
## [[Motivation and emotion/Book/2010/Hunger motivation|Hunger motivation]] - [[User:JWSchmidt|JWSchmidt]]
## [[Motivation and emotion/Textbook/Motivation/Sexual motivation|Sexual motivation]] - [[User:Dos|Dos]]
### [[Motivation and emotion/Textbook/Motivation/Adultery|Adultery]] - [[User:S.emp|S.emp]]
### [[Motivation and emotion/Textbook/Motivation/Alcohol consumption and sexual motivation|Alcohol consumption and sexual motivation]] - [[User:Ra.shell|Ra.shell]]
### [[Motivation and emotion/Textbook/Motivation/Gender differences in sexual motivation|Gender differences in sexual motivation]] - [[User:Jasmine13|Jasmine13]]
### [[Motivation and emotion/Textbook/Motivation/Mate-seeking behaviour|Mate-seeking behaviour]]  - [[User:Klammer|Klammer]]
### [[Motivation and emotion/Book/2010/Promiscuity motivation|Promiscuity motivation]] - [[User:Lmckenz|Lmckenz]]
### [[Motivation and emotion/Textbook/Motivation/Sex offenders|Sex offenders]] - [[User:Esha434|Emmarie]]
# Mini-theories
## [[Motivation and emotion/Textbook/Motivation/Arousal|Arousal]] - [[User:E.herbert|E.herbert]]
## [[Motivation and emotion/Textbook/Motivation/Flow|Flow theory]] - [[User:KmpH|KmpH]]
## [[Motivation and emotion/Textbook/Motivation/Learned helplessness|Learned helplessness]] - [[User: JackStraw| Jack Straw]]
# Motivation and behaviour
## [[Motivation and emotion/Textbook/Motivation/Dieting|Dieting]] - [[User:Christie88|Christie88]]
## [[Motivation and emotion/Textbook/Motivation/Exercise|Exercise motivation]] - [[User:Haddo|Haddo]]
## [[Motivation and emotion/Textbook/Motivation/Gambling|Gambling]] - [[User:Matt.long|Matt.long]]
## [[Motivation and emotion/Book/2010/Motivation and goal setting|Motivation and goal setting]] - [[User:MissadventureX|MissadventureX]]
## [[Motivation and emotion/Book/2010/Overeating motivation|Overeating motivation]] - [[User:Madeline|Madeline]]
## [[Motivation and emotion/Book/2010/Procrastination|Procrastination]] - [[User:Sallybradford|Sallybradford]]
## [[Motivation and emotion/Textbook/Motivation/Risk-taking|Risk-taking]] - [[User:AlEdwardson|AlEdwardson]]
## [[Motivation and emotion/Textbook/Motivation/Social inhibition|Social inhibition]]  - [[User:Harro242|Harro242]]
## [[Motivation and emotion/Textbook/Motivation/Violence|Violence]] - [[User:Spartan117|Spartan117]]
# Motivation and cognition
## [[Motivation and emotion/Textbook/Motivation/Positive thinking|Positive thinking]] - [[User:Rachelle21|Rachelle21]]
# Motivation and culture
## [[Motivation and emotion/Book/2010/Indigenous Australians and motivation|Indigenous Australians and motivation]] - [[User:Mish795|Mish795]]
# Motivation and education
## [[Motivation and emotion/Textbook/Motivation/Education|Motivation and education]] - [[User:U3005872]]
## [[Motivation and emotion/Book/2010/Student motivation theories|Student motivation theories]] - [[User:U118827|U118827]]
# Motivation and psychological disorders
## [[Motivation and emotion/Textbook/Motivation/Anxiety|Anxiety]] - [[User:Hamish24|Hamish24]]
## [[Motivation and emotion/Textbook/Motivation/Antisocial personality disorder|Antisocial personality disorder]] - [[User:AriWright|AriWright]]
## [[Motivation and emotion/Book/2010/Dementia and motivation|Dementia and motivation]] - [[User:Skye.marie|Skye.marie]] 
## [[Motivation and emotion/Book/2010/Depression and motivation|Depression and motivation]] - [[User:Megan O'Connell|Megan O'Connell]]
## [[Motivation and emotion/Book/2010/Narcissism|Narcissism]] - [[User:Aszokalski|Aszokalski]]
## [[Motivation and emotion/Textbook/Motivation/Narcissism2|Narcissism]] - [[User:A Hock|A Hock]]
## [[Motivation and emotion/Textbook/Motivation/Paraphilias|Paraphilias]] - [[User:CEB|CEB]]
# Neurobiology of motivation
## [[Motivation and emotion/Book/2010/Motivational toxicity|Motivational toxicity]] - [[User:Mylie|Mylie]]
# Motivation and self
## [[Motivation and emotion/Book/2010/Self-concept|Self-concept]] - [[User:Boubles|Boubles]]
## [[Motivation and emotion/Textbook/Motivation/Self-discipline|Self-discipline]] - [[User:Mike.j|Mike.j]]
## [[Motivation and emotion/Book/2010/Self-sabotage motivation|Self-sabotage motivation]] - [[User:Bails|Bails]]
# [[Motivation and emotion/Book/2010/Personality and motivation|Personality and motivation]] - [[User:George902|George902]]
# Types of motivation
## [[Motivation and emotion/Textbook/Motivation/Achievement motivation|Achievement motivation]] - [[User:Gabrielleblair|Gabrielleblair]]
### [[Motivation and emotion/Textbook/Motivation/Aggression/Workplace|Aggression in the workplace]] - [[User:Salbo|Salbo]]
## [[Motivation and emotion/Textbook/Motivation/Self-actualisation|Self-actualisation]] - [[User:Dan.th.man|Dan.th.man]]
## [[Motivation and emotion/Textbook/Motivation/Spiritual|Spiritual]] - [[User:Dchimself|Dchimself]]
# [[Motivation and emotion/Book/2010/Unconscious motivation|Unconscious motivation]] - [[User:Katrina3027900|Katrina3027900]]

==Emotion==
[[File:Love's Passing.jpg|right|160px]]
[[File:Naya, Carlo (1816-1882) - n. 553a - Carpaccio V. 1506 - Dettaglio del sogno di Santa Orsola (La testa della Santa) - Academia, Venezia.jpg|right|160px]]
# Aspects of emotion
## [[Motivation and emotion/Textbook/Emotion/Emotional expression|Emotional expression]] - [[User:Shannonld|Shannonnld]]
## [[Motivation and emotion/Textbook/Emotion/Emotion management|Emotion management]] - [[User:Jmcb|Jmcb]]
## [[Motivation and emotion/Textbook/Emotion/Emotional stability|Emotional stability/instability]] - [[User:Clinton.mcculloch|Clinton.mcculloch]]
# Basic/Core emotions
## [[Motivation and emotion/Book/2010/Ekman's basic emotions|Ekman's basic emotions]] - [[User:MichelleK|MichelleK]]
# Emotion and behaviour
## [[Motivation and emotion/Textbook/Emotion/Facial expression|Facial expression]] - [[User:Tink22|Tink22]]
# Emotion and psychological disorders
## [[Motivation and emotion/Textbook/Motivation and emotion/Textbook/Emotion/Antisocial personality disorder|Antisocial personality disorder]] - [[User:Emma22|Emma22]] 
## [[Motivation and emotion/Textbook/Emotion/Psychopathy|Psychopathy]] - [[User:Lisa Watson|Lisa Watson]]
# Emotion and specific groups
## [[Motivation and emotion/Textbook/Emotion/Adolescence|Adolescence]] - [[User:Stephmartin|Stephmartin]]
# Emotion and specific topics
## [[Motivation and emotion/Textbook/Emotion/Culture|Culture]] - [[User:FilL|FilL]]
## [[Motivation and emotion/Textbook/Emotion/Sport|Sport]] - [[User:Lozh|Lozh]] 
## [[Motivation and emotion/Textbook/Emotion/Music|Music]] - [[User:Rotorhead15|Rotorhead15]]
## [[Motivation and emotion/Textbook/Emotion/Sex|Sex]] - [[User:Nikki24|Nikki24]]
## [[Motivation and emotion/Book/2010/Sleep and emotion|Sleep]] - [[User:Storm|Storm]]
## [[Motivation and emotion/Book/2010/Emotional development in children|Emotional development in children]] - [[User:WMLee|WMLee]]
# [[Motivation and emotion/Textbook/Emotion/Mood|Mood]] - [[User:LBGibbons|LBGibbons]]
# Specific emotions
## [[Motivation and emotion/Textbook/Emotion/Neurobiology of aggression|Neurobiology of aggression]] - [[User:u3006008|u3006008]]
## [[Motivation and emotion/Textbook/Emotion/Anxiety|Anxiety]] - [[User:Gajah|Gajah]]
## [[Motivation and emotion/Book/2010/Empathy|Empathy]] - [[User:JennyJ|JennyJ]]
## [[Motivation and emotion/Textbook/Emotion/Happiness|Happiness]] - [[User:SandyC|SandyC]]
# [[Motivation and emotion/Textbook/Emotion/Stress, arousal and coping|Stress, arousal and coping]] - [[User:Barbie|Barbie]]
# [[Motivation and emotion/Textbook/Emotion/Stress and health|Stress and health]] - [[User:C.aitken|C.aitken]]
# [[Motivation and emotion/Textbook/Emotion/Theories/Cognitive|Cognitive theories of emotion]] - [[User:U3017048|U3017048]]

==Motivation &amp; emotion==
# [[Motivation and emotion/Textbook/Motivation and emotion/Animals|Motivation and emotion in animals]] - [[User:Mixie|Mixie]]

==Summary==
# [[Motivation and emotion/Textbook/Summary and conclusion|Summary and conclusion]] - [[User:Jtneill|Jtneill]]
# [[Motivation and emotion/Textbook/Feedback|Feedback]]

==See also==
* [[Motivation and emotion/Assessment/Chapter|Textbook chapter guidelines]]
* [[Motivation and emotion/Textbook/Structure|Textbook structure and features]]
* [[Motivation and emotion/Textbook/Pedagogy|Textbook pedagogy]]
* [[b:Cognitive Psychology and Cognitive Neuroscience/Motivation and Emotion|Cognitive Psychology and Cognitive Neuroscience/Motivation and Emotion]] (Cognitive psychology and cognitive neuroscience (Wikibooks))

==External links==
* [http://www.google.com/search?tbs=bks%3A1&amp;tbo=1&amp;q=motivation&amp;btnG=Search+Books Motivation] (Google Books)
* [http://www.google.com/search?hl=en&amp;safe=off&amp;tbo=1&amp;tbs=bks%3A1&amp;q=emotion&amp;aq=f&amp;aqi=g10&amp;aql=&amp;oq=&amp;gs_rfai= Emotion] (Google Books)
* [http://www.springer.com/psychology/journal/11031 Motivation and Emotion] (Journal) - [http://www.springerlink.com/content/h565j2qtl1p0/ Table of contents]
[[Category:Motivation and emotion/Textbook| ]]
[[Category:User:Jtneill/Grants/Student-authored open textbooks 2010]]
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==Overview==
[[File:Fingerprint picture.svg|thumb|right|250px|Self-Concept: The unification of self-schemas to form a sense of self]]
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{{big2|'''Box 1.1 Self-Concept of a Boy Aged 9'''}}&lt;br&gt;
[[File:Ball balance.jpg|center|100px]]
My name is Bruce C. I have brown eyes. I have brown hair. I have brown eyebrows.
I am 9 years old. I LOVE sports. I have 7 people in my family. I'm a boy. I LOVE food.
&lt;small&gt;(Adapted from Ross, 1992)&lt;/small&gt;
|}

'''Who are you?''' Are you tall? Short? Do you have brown or blonde hair? Is academic competence important to you? Do you only value knowledge and intelligence in a particular domain, say, maths or history? Are you a fit and althetic person? Do you get along with your peers or work colleagues? Are you shy or boisterous?

Each of these questions reflects a particular self-schema which forms the framework of one's self-concept. Self-concept can be understood as guiding the energy and direction behind [[attitude]]s, [[emotion]]s and [[behaviour]]s. Why do you dress the way you do? Why did you enrol in university or apply for a particular job? An individual's self-concept permeates all facets of his/her life and is a key motivational factor underlying attitudes, emotions and behaviours in the hope to attain facets of [[psychological well-being]] (Ross, 1992).

==={{font|color=Purple|What is Self-Concept and Motivation and how are they Related?}}===

'''Self-concept'''

''''Self'''' as a noun came into the English language around AD 1400 and was initially defined by negative connotations, such as selfishness (Ross, 1992). The negative connotations reflect the historical context as seen in the following pledge; ''&quot;Our own self we shall deny, and follow our Lord Almighty&quot;'' (Baumeister, 1986).

This trend continued into the 16th century when hyphenations of the self became popularised, such as self-pity, self-praise and self-conceit. From the 17th century onwards, the self took on a more positive light with the development of terms such as self-interest, self-efficacy and self-determination (Ross, 1992). The importance and purpose behind the self also shifted in direction as portrayed by the following 18th century Nathaniel Cotton poem excerpt; ''&quot;The world has nothing to bestow; From our own selves our joys must flow&quot;'' (Cotton, 2003).

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{{big2|'''Box 1.2 Self-Concept of a Girl Aged 11 1/2'''}}&lt;br&gt;

My name is A. I'm a girl. I'm not pretty. I do so-so in my studies. I'm a very good pianist.
I'm old-fashioned. Mostly I'm good, but I lose my temper. I'm not well-liked by some girls and boys.
&lt;small&gt;(Adapted from Ross, 1992)&lt;/small&gt;
|}

The term '''self-concept''' emanates from the previously mentioned derivatives of the self, for example, it incorporates aspects of self-interest, self-praise and self-efficacy. Specifically, self-concept can be understood as one's conception of themselves as a distinct individual; mental representations of who one is and who they wish to become in the context of their environment (Beck, 2000). Box 1.1, 1.2 and 1.3 provide examples of the developing self-concepts of three young people. Reeve (2009) asserts that the self-concept develops from personal experiences, reflections on these experiences and feedback from the social environment. Thus, the process of self-concept development and consolidation involves a reciprocal, cyclic process as depicted in the following diagram:

[[File:Reciprocal self-concept.JPG|center|350px]]

The self-concept is organised into a semantic hierarchy of '''self-schemas'''; cognitive generalisations which are domain specific (Reeve, 2009). Self-schemas can include appraisals in social (peers, significant others), academic (general or specific intelligence), emotional (specific emotions), or physical (abilities, appearance) domains (Ross, 1992). Within the self-schema hierarchy, individuals can also possess high or low levels of differentiation and integration (Lipka &amp; Brinthaupt, 1992).

'''Differentiation''' refers to the breadth of an individual's self-concept with children often exhibiting low differentiation and adults high differentiation. For example, a child's hierarchy may revolve around a limited number of self-schemas, such as physical ('I am tall', 'I have brown hair') and academic ('I love math', 'I dislike English') domains. Conversely, adults often possess more extensive hierarchies, incorporating many domains (Ross, 1992).

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{{big2|'''Box 1.3 Self-Concept of a Girl Aged 17'''}}&lt;br&gt;
[[File:Abigail Hyndman, 15-year-old from Bodelwyddan.jpg|center|120px]]
I am a human being. I am a girl. I am an individual. I don't know who I am.
I am moody, indecisive and ambitious. I am not an individual. I am not a classifiable person (i.e. I don't want to be).
&lt;small&gt;(Adapted from Ross, 1992)&lt;/small&gt;
|}

'''Integration''' subsequently defines how cohesive and interrelated an individual's hierarchy is, with children again showing low levels and adults high levels. For example, a child often displays disjointed self-schemas such as defining themselves by hair colour (physical domain) and athletic ability (physical domain). Alternatively, adults often define themselves through an integrated network of self-schemas such as their social abilities (social domain) based on interactions with family, peers, strangers and work colleagues and how this relates to their emotional intelligence (academic domain) and the way they present themselves (physical domain) (Deckers, 2004).

'''Motivation'''

[[Motivation]] refers to the energising and directive properties behind human behaviour which determines the quantity and quality of our actions (Reeve, 2009). It is shaped and driven by an individual's unique composition internal and external motives including biological, physiological, psychological, cognitive and environmental forces (Beck, 2000). Motivation energises and directs an individual to work out who they are (self-assessment), create a consistent and accurate hierarchy of self-schemas (self-verification) and enable control and psychological well-being (self-enhancement). The purpose of the self-concept therefore reflects the human tendency to seek consistency, control and predictability over their attitudes, emotions and behaviours which will be discussed later in the attribution theory (Lipka &amp; Brinthaupt, 1992).

==={{font|color=Purple|Chapter Outline and Focus Questions}}===

* '''Introduction'''
** How is self-concept and motivation defined?
** What is the relationship between self-concept and motivation?
** What is the importance of this topic?

* '''Self-concept development'''
** How do biological, brain structure, socio-cultural and psychological elements contribute to self-concept development?

* '''Self-concept stability'''
** How do human tendencies and motives influence self-concept development?
** How stable is the self-concept?
** Why is self-concept stability important?

* '''Self-concept change'''
** What causes self-concept change?
** How is the self-concept changed/managed?
** When are individuals most aware of undesirable self-concepts?
** Can we become the person we want to be?
** How does social comparison affect the self-concept?

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== Self-Concept Development ==

[[File:Aging faces manitou2121.jpg|thumb|200px|left|Physical attractiveness influences an individual's self-concept..]]

''Self-concept development is influenced by numerous factors including biological, brain structure, neurotransmitter, socio-cultural and psychological elements. Each element has a differential quantity and quality of impact on an individual and combined they create a unique self-concept. Consequently, individuals demonstrate differences in motivated attitudes, emotions and behaviour which produces varying levels of self-concept change, management and stability (Deckers, 2004).''

==={{font|color=Purple|Biological Elements}}===

Biological elements, including genotype and phenotype, play an important role in the development and maintenance of an individual's self-concept (Ross, 1992). The genotype describes the internal genetic code which uniquely controls the inner-workings of each individual. The primary link between genotype and self-concept is the ability for the genotype to control the phenotype. The phenotype defines the outward manifestation of an individual via physical appearance, personality and behaviour (Maltby, Day &amp; Macaskill, 2007). Thus, genetics influence the development of particular self-schemas, such as in social, physical or academic domains, which interact to form an individual’s self-concept.

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{{big2|'''Box 1.1 Big-5 Personality Traits'''}}&lt;br&gt;
* Openness: curious, analytical, perceptive, knowledgeable
* Conscientiousness: cautious, reliable, organised, hardworking
* Extraversion: sociable, talkative, enthusiastic
* Agreeableness: warm, cooperative, trustful
* Neuroticism: anxiety, depressive, self-conscious, emotional
&lt;small&gt;(Maltby, Day &amp; Macaskill, 2007)&lt;/small&gt;
|}

Physical attractiveness has firstly been shown to influence an individual’s self-concept. Adams and Read (1983) demonstrated that attractive individuals perceived themselves as possessing significantly more positive personality traits than unattractive individuals, such as a superior analytical and critical thought ability and a warm, extraverted social manner. Conversely, unattractive participants perceived themselves as possessing more negative than positive qualities, such as poorer control of interpersonal situations. Similarly, Marks, Miller and Maruyama (1981) found participants were biased when rating others, for example, attractive samples were deemed more intelligent, thoughtful and open-minded than unattractive samples. Interestingly, even psychological therapists have been found to exhibit this bias when evaluating a new client (Hobfoll &amp; Penner, 1978).

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{{big2|'''Intelligence and Self-Concept'''}}&lt;br&gt;

Intelligence also plays a role in self-concept development, such as shaping social, physical and academic-domain self-schemas..
&lt;small&gt;(Maltby, Day &amp; Macaskill, 2007)&lt;/small&gt;
|}

Physical attractive bias does appear to partly comprise a biological component as Langlois, Roggman and Rieser-Danner (1990) demonstrated that infants displayed positive affective tone, less withdrawal behaviour and high play involvement with an attractive confederate or doll as compared to an unattractive confederate or doll. Socio-cultural elements tend to stimulate and consolidate the presence of physical attractive biases. For instance, Salmivalli (1998) found school yard bullies often contained a high but negative social and physical self-concept and victims a low social and physical self-concept. Bias is also evident within occupational domains with attractiveness positively correlated with higher income (Judge, Hurst &amp; Simon, 2009), flexible benefits and individual-based pay rates (Cable &amp; Judge, 1994). Individuals also tend to engage in romantic relationships of similar physical attractiveness which is argued to partly reflect an evolutionary basis to maintain survival (which in contemporary society can relate to status and wealth) and reproduction (Horton, 2003). Thus, attractiveness influences how an individual perceives themselves in addition to how others perceive them.  The consequences of this bias shapes self-concept development and maintenance. At its most extreme, attractiveness bias can lead individuals to unrealistically distort or evaluate their physical self-schema which plays a role in the development of eating disorders (Jansen, Smeets, Martijn &amp; Nederkoorn, 2006).

[[Personality]] traits also influence an individual’s self-concept as they energise and direct approach or avoidance behaviour toward particular self-schemas (Maltby et al., 2007). Twin, adoption and family study research indicates personality traits are partly hereditary, accounting for approximately 20-50% of the variance in self-concept (Lounsbury, Levy, Leong &amp; Gibson, 2007). The big-5 theory of personality provides a common framework for understanding hereditable traits, as displayed in Box 1.1. McCroskey, Heisel and Richmond (2001) demonstrated that extraverts were more assertive and self-accepting than introverts and tended to approach social and arousing occupations and relationships. Conversely, introverts have a tendency to be socially inhibited and apprehensive, preferring predictable and independent occupations and relationships. van der Zee, Thijs and Schakel (2002) found neuroticism to be negatively correlated with [[emotional intelligence]] which subsequently appeared to weaken academic intelligence and research on interpersonal conflict suggests individuals high on agreeableness are more responsive to conflict and enact strategies to diffuse these situations (Jensen-Campbell &amp; Graziano, 2001).  Research therefore indicates that personality traits predispose individuals with certain abilities and tendency which subsequently influence self-concept development.

==={{font|color=Purple|Brain Structure Elements}}===

[[File:Brain limbicsystem.svg|thumb|200px|left|&lt;big&gt;'''Mood &amp; Self-Concept:''' A positive mood is often associated with a more positive self-concept, while a negative mood is often associated with a more negative self-concept&lt;/big&gt; (Showers, Abramson &amp; Hogan, 1998)]]

Specific brain structures influence self-concept development, depending on individual differences, which can be illustrated through Eysenck's biological model of personality and arousal (Lounsbury et al., 2007). The model asserts that extraverted and neurotic personality dispositions trigger specific brain structure activity which influences behaviour and self-schema development. The model firstly claims that individuals attempt to maintain an equilibrium between their excitatory (alert, active and aroused) and inhibitory (inactivity and lethargy) mechanisms. Equilibrium is achieved through the ascending reticular activating system (ARAS) which controls arousal via the reticulo-cortical circuit (incoming stimuli) and reticulo-limbi circuit (emotional stimuli). The ARAS is situated in the brain stem and connects the thalamus (switchboard), hypothalamus (metabolism &amp; autonomic processes) and cortex (neural processing) to provide an integrated response to information and stimulation (Maltby et al., 2007).

The second component of the model relates to individual differences based on extraverted and neurotic personality types. Extraversion is associated with arousal of the reticulo-cortical (incoming stimuli) circuit in which low arousal levels are present in extraverts (as they are under-aroused and desire stimulation) and high arousal levels are present in introverts (as they are over-aroused and attempt to avoid stimulation). This assertion is supported by literature mentioned in the biological section, such as extraverts preferring social and arousing environments and introverts preferring independent and predictable environments (McCroskey et al., 2001). Alternatively, neuroticism is associated with the arousal of the reticulo-limbic (emotional stimuli) circuit in which neurotics tend to be more aroused by emotional stimulation than stable individuals. Again, this is reflected in aforementioned literature such as neurotics' inability to demonstrate emotional intelligence (van der Zee et al., 2002).

==={{font|color=Purple|Socio-cultural Elements}}===

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{{big2|'''Culture and Self-Concept'''}}&lt;br&gt;
Culture can influence self-concept, for example, Western societies promote values of autonomy and individuality. Conversely, the Ilongots from the Philippines value social similarity and collectivism.
&lt;small&gt;(Brinthaupt &amp; Lipka, 1992)&lt;/small&gt;
[[File:Guillaume Bastille.jpg|center|150px]][[File:Tinguin men of Sallapadin.jpg|200px]]
|}

Socio-cultural elements play a role in self-concept development including the influence of [[gender role]]s, family, peers and [[religion]]. Research suggests that most cultures socialise individuals into particular gender roles, for example, Western society overtly and covertly encourages femininity amongst females and masculinity amongst males which continues throughout the life span (Gouze &amp; Nadelman, 1980). Bassen and Lamb (2006) posit that gender roles sway the development of schemas in social, emotional and academic domains. For example, male participants frequently indicated their social and emotional self-schema comprised qualities such as assertiveness whereas females highlighted qualities such as affiliativeness. Furthermore, Rudman and Phelan's (2010) study demonstrated that priming females with traditional gender [[stereotypes]] (female: teacher, male: mechanic) led to less reported interest in male-stereotyped roles. Interestingly, priming of nontraditional roles (female: mechanic, male: teacher) resulted in participants reporting a lowered leadership self-schema because they engaged in upward social comparison and therefore perceived the scenario as a threat. Gender stereotype threat is a final example of how prescribed roles influence self-concept. A common example in literature is the proposal that males perform better than females on maths equations. This subsequently lowers female performance and academic-domain self-schemas due to performance pressures. Importantly, this effect appears to be limited to females who report a high-identification academic-domain schema (Keller, 2007).

Family and peer interactions also shape an individual's self-concept. Triadic family interaction refers to the social relations between mother, father and child (Brown, Mangelsdorf, Neff, Schoppe-Sullivan &amp; Frosch, 2009). The family systems' perspective posits that the attributes and behaviours of family members, parenting techniques and the differential roles of the mother and father influences self-concept development in childhood (Brown et al., 2009; von Wyl et al., 2008). For instance, Brown et al. (2009) found harmonious interactions were correlated with children expressing positive self-schemas, such as adventurousness. Conversely, discordant interactions such as hostility or low engagement were associated with self-schemas including fearfulness and less agreeableness. Thus, a child is likely to experience constructive self-concept development if they are exposed to parents who (a) possess supportive attributes or (b) provide positive role-modelling or (c) utilise supportive and encouraging parenting techniques or (d) fulfil nurturing mother or father roles. Similarly, supportive or aversive peer interactions shape an individual's self-concept, particularly during middle childhood and adolescence (Lipka &amp; Brinthaupt, 1992). For example, Egbochuku (2009) found that secondary school females reported higher self-concept in academic and social domains when they attended single-sex as opposed to co-educational schools. This finding was explained by the impact of differing peer interactions and subsequent influences.

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{{big2|'''Relgion, Social Class and Self-Concept Study'''}}&lt;br&gt;
[[File:Catholic Nun.svg|80px|center]]
This particular study found:
* Catholics possessed higher 'love' scores than Jewish participants
* Upper class participants demonstrated higher 'dominance' scores than lower class participants
* These qualities influenced participants social self-schemas and general self-concept
&lt;small&gt;(Bieri &amp; Lobeck, 1961)&lt;/small&gt;
|}

Lastly, the [[media]] also contributes to self-concept development. The level of influence depends on the extent to which an individual attributes importance to the medium and subsequently internalises the content (Beck, 2000). The media exerts influence in most self-schema domains including social, academic, emotional, physical and sexual. For example, Aubrey (2007) found female university students developed negative and dissatisfied sexual self-schemas from viewing soap operas, prime time dramas or excessive amounts of television. Similarly, Bessenoff (2006) found excessive television, newspaper or magazine viewing was associated with the exacerbation of actual-ideal physical self-schemas, particularly for women and thin body shapes. However, the media is also claimed to produce positive self-concept development, such as providing education outlets which develops an individual's academic self-schema (Ross, 1992).

==={{font|color=Purple|Psychological Elements}}===

The development of an individual's self-concept is also shaped by psychological elements which gain increasing complexity throughout the life span. A famous experiment demonstrated the ability of infants aged 12 to 18 months to engage in self-recognition through identifying rouge on their nose when looking at a mirror (Lipka &amp; Brinthaupt, 1992). This illustrates an individual's first experience of identifying the existential self. Infants then develop self-awareness through self-perception exercises, for example, kicking a toy mobile is connected to the mobile swinging and making sounds. Subsequently, the swinging and sounds are associated with the emotion of joy. The ability to connect events and emotions therefore develops an individual's initial journey to understanding their likes and dislikes which directs their behaviour in later life (Ross, 1992).

Attachment behaviour during early childhood marks the ability to distinguish humans and develop one's social self-schema. Research suggests three attachment styles including secure (adventurous, healthy child-mother attachment), resistance (afraid of solo play, crying upon reunion with mother) and avoidant (sporadic play, avoidance upon reunion). These attachment styles are subsequently claimed to continue into middle childhood and influence an individual's understanding of their self-concept (Beck, 2000).

Throughout childhood and adolescence, the acquirement of language and learning of cognitive labels plays the key role in establishing a concrete understanding of one's self-concept. Research shows that during this period individuals learn to distinguish between purely external qualities, such as appearance and possessions, and internal qualities, such as personality traits and possible selves (Salmivalli, 1998). During adulthood psychological needs including autonomy, competence and relatedness motivate the finer distinctions within one's self-concept. For example, achieving a goal under autonomy-supportive conditions, rather than controlled regulation, has been associated with eudaimonic self-concept development (Ryan &amp; Deci, 2000).

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== Self-Concept Stability ==

''Two universal factors which influence self-concept development and consolidation are human tendencies, outlined by the attribution theory, and human motives, including self-assessment, self-verification and self-enhancement. These tendencies and motives shape and predict self-concept-related attitudes, emotions and behaviours in addition to change, management and stability. The underlying purpose of these tendencies and motives is to ensure control, consistency, predictability and psychological well-being.''

[[File:Example of Weiner at school.pdf|600px|center|thumb|'''Figure 1. Differing academic self-schemas as determined by attribution type''']]

==={{font|color=Purple|Attribution Theory}}===

The '''attribution theory''' asserts that humans have a tendency to attribute causality to events, people and situations (West &amp; Turner, 2007). Individuals commonly display three inferences; internal or external, stability and controllability attributions. An internal attribution infers an internal cause, such as explaining attitudes, emotions and behaviour by personality, mood, ability or effort. Conversely, external attributions infer an external cause, such as luck, social influence or task difficulty. Stability describes the attribution of a stable or temporary cause, such as personality dispositions or the effects of alcohol consumption. Controllability is a final attribution which is the distinction between a controllable or uncontrollable cause, such as a situation which was influenced by effort or luck. Figure 1 outlines differing attributions for an individual's academic self-schema. This behaviour enables an individual to develop a coherent view of the world and maintain some level of control, consistency and predictability (Graham &amp; Folkes, 1990).

[[File:Mirror baby.jpg|200px|left|thumb|'''Self-assessment: getting to know who you truly are''']]

==={{font|color=Purple|Self-Assessment, Self-Verification and Self-Enhancement}}===

The attribution theory therefore provides the basis for motivated behaviour and self-assessment, verification and enhancement build upon this basis to shape and predict self-concept-related attitudes, emotions and behaviours. '''Self-assessment''' defines the desire to gain an accurate perception of one's self-concept, regardless of negative or positive findings (Crisp &amp; Turner, 2010). This desire relates back to the attribution theory as individuals seek to reduce uncertainty and gain predictability over their attitudes, emotions and behaviour by possessing accurate self-concept knowledge. '''Self-verification''' subsequently refers to the desire to confirm and consolidate our self-concept. Similarly, this motive derives from the desire to gain control, consistency and predictability over one's self-concept. '''Self-enhancement''', unlike the former two motives, specifically aims to enable positive perceptions about one's self-concept. This is argued to be the most influential motive and often overrides the former two motives because it maintains self-concept stability and psychological well-being through high self-esteem (Elliott, 2007).

[[File:Winning the Gold!.jpg|200px|left|thumb|'''Self-verification: confirming what we already believe about ourselves''']]

==={{font|color=Purple|How do Human Tendencies and Motives Influence the Self-Concept?}}===

It is therefore important to keep in mind that self-concept development, change, management and stability stem from the underlying need to attribute causality and maintain control, consistency and predictability. Human tendencies and motives influence self-concept development as individuals innately seek out information to understand and confirm and enhance their perceptions. For example, gender roles provide a way for individuals to carry out self-assessment, social interaction enables self-verification and psychological processes such as [[self-serving bias]] promote self-enhancement.

The next section addresses precursors for self-concept change and management, which is also influenced by human tendencies and motives. Inconsistent attitudes and behaviours (cognitive dissonance theory), actual-ideal or actual-ought selves (self-discrepancy theory) and negative social comparisons (self-evaluation maintenance model and social identity theory) all impinge on an individual's sense of control, consistency, predicatability and psychological well-being (Pemberton &amp; Sedikides, 2001). Thus, a fundamental motivation within self-concept change and management involves the recognition of undesirable states and subsequent efforts to restore stable, accurate and positive self-schemas.

==={{font|color=Purple|How Stable is the Self-Concept?}}===

Research suggests that self-concept is both stable and malleable (Lipka &amp; Brinthaupt, 1992). An individual often possesses a core sense of self which is complemented by an ever-changing outer self. The core self comprises biological elements and innate human tendencies and motives. For example, an individual's genotype and phenotype predisposes them with a particular gender role and unique personality traits, temperaments and neural functioning. Furthermore, the aforementioned human tendencies and motives encourage individuals to constantly seek out and identify a controllable, consistent and predictable self-concept.

The ever-changing outer self-concept is influenced chiefly by psychological and socio-cultural elements. Psychological influences include the recognition of unstable self-schemas (cognitive dissonance) and discrepancies within ideal and possible selves (self-discrepancies). Socio-cultural influences include factors such as peer or family relations, religious beliefs, social class, cultural background or the effects of social comparison (self-evaluation maintenance model and social identity theory). Thus, each of these elements play a role in modifying and altering an individual's self-concept.

==={{font|color=Purple|Why is Self-Concept Stability Important?}}===

A minimum level of self-concept stability is important for psychological well-being and optimum functioning (Elliott, 2007). The importance of stability can be understood through the [[Wikipedia:Self-determination theory#Basic needs and intrinsic motivation|self-determination theory]]. The theory asserts that optimal motivation stems from humans' inherent growth tendencies and innate psychological needs. Specifically, motivation falls along a continuum of perceived locus of causality which ranges from extrinsic motivations (e.g. external regulation and compliance) to intrinsic motivations (e.g. intrinsic regulation and personal interest and agency) (Deckers, 2004). A individual's locus of causality subsequently derives from the extent of psychological need satisfaction, including autonomy, competence and relatedness needs. The ability to equally fulfil these psychological needs and pursue self-concordant goals promotes intrinsic motivation which fosters self-concept stability and psychological well-being, such as self-acceptance, positive interpersonal relations, autonomy, environmental mastery, purpose in life and personal growth (Reeve, 2009).  Table 1 outlines specific facets of psychological well-being obtained from a stable and well-developed self-concept.

'''Table 1. The Importance of Self-Concept Stability and its Relationship to Self-Determination Theory'''
{| border=1 cellspacing=0 cellpadding=5
|-
! Facet of Psychological Well-Being
! Example
|-
| 1. Self-acceptance: positive self-esteem and acceptance of positive and negative self-schemas and past experiences
| Slight, gradual change is often a natural process for self-acceptance and self-concept stability which is often demonstrated through the transition from adolescence to adulthood  &lt;small&gt;(Salmivalli, 1998).&lt;/small&gt;
|-
| 2. Positive interpersonal relations: the ability to form and maintain intimate, empathetic and deep relationships
| [[Wikipedia:Self-control|Self-regulation]] promotes selective interaction which is the process of selecting friends who will provide effective social support and share similar values which reinforces an individual's self-concept and aids in self-enhancement through positive social feedback &lt;small&gt;(Egbochuku, 2009).&lt;/small&gt;
|-
| 3. Autonomy: fulfilling psychological needs to obtain intrinsic motivation toward self-concordant goals
| Successful athletes often possess social support, agency in pursuing their sporting goals and competency in their physical domain which produces intrinsic motivation and effective performance &lt;small&gt;(Thrash &amp; Elliot, 2002).&lt;/small&gt;
|-
| 4. Environmental mastery: ability to seek out and interact effectively with optimal environments	
| The ability to demonstrate self-awareness and perception to gain information about the self-concept (self-assessment), use this information to confirm positive self-schemas (self-verification) and promote self-efficacy in domain-related tasks (self-enhancement) &lt;small&gt;(Roney &amp; Sorrentino, 1995).&lt;/small&gt;
|-
| 5. Purpose in life: the development of meaning and direction in life	
| The ability to assess, verify and enhance the self-concept is important for the development of meaning and direction in life, for example, the formation of ideal and positive selves as well as self-concordant goals &lt;small&gt;(McDaniel &amp; Grice, 2008).&lt;/small&gt;
|-
| 6. Personal growth: self-concept improvement, growth and actualisation
| An individual who can optimally integrate the former facets of psychological well-being place themselves in a position to move beyond hedonic well-being (pleasure attainment, pain avoidance) to the attainment of eudaimonic well-being (self-concept actualisation, fully functioning) &lt;small&gt;(Ryan &amp; Deci, 2001).&lt;/small&gt;
|-
|}

''{{font|color=Purple|Summary:}} Self-concept comprises a core part and ever-changing outer part which interact to produce motivated attitudes, emotions and behaviours. Change is often necessary for self-concept development and stability as it enables an individual to assess, modify and reaffirm their self-schemas. Furthermore, change often takes the form of only slight, gradual differences which enables the core part to maintain effective functioning. Although a lack of self-awareness can lead to the pursuance of self-disconcordant goals, humans possess an innate tendency to strive for self-concordant goals. This enables the fulfilment of psychological needs, such as striving for autonomy through life goals, competence through environmental mastery and relatedness through positive interpersonal relations. Together, self-concordant goals and psychological need satisfaction encourage the development of intrinsic motivation which enables persistence, creativity, conceptual understanding and optimal functioning (Reeve, 2009).''
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== Self-Concept Change and Management ==

''The final section of this chapter addresses specific forms of self-concept change and management. As previously mentioned, self-concept change often takes the form of slight, gradual differences and to a limited number of self-schema domains at one point in time. As self-concept construction is deemed to stem from personal reflections on experiences as well as social feedback, three key theories will be addressed. The cognitive dissonance theory firstly outlines self-concept change derived from personal reflections on experiences and subsequent motivational properties designed to obtain attitude and behaviour equilibrium. Secondly, the self-discrepancy theory highlights how personal reflection can trigger awareness of discrepancies between actual, ideal and ought selves in addition to possible selves. Positive and negative self-esteem is also addressed to illustrate its role in preventing or enabling a person to become who they want to be. The final theory, self-evaluation maintenance model, highlights the influence of social feedback on self-concept change and management. It further emphasises the motivational properties produced through social comparison.''

==={{font|color=Purple|Cognitive Dissonance Theory}}===

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'''Cognitive Dissonance: In Focus'''

This case study follows Liz and Ben, a once happy couple who are now experiencing cognitive dissonance because their attitude ('we love each other') is inconsistent with their behaviour (constant bickering).

'''Dissonance-arousing Situations:'''
* Choice: they have chosen to maintain their own sides of the argument. Liz recognises that her attitude ('happy couples compromise') is inconsistent with her behaviour ('sticking to her opinion').
* Insufficient justification: Liz realises that she automatically rejects a call from Ben which makes her uncomfortable because she believes committed couples will always display an implicit desire to resolve conflict. She subsequently appraises the situation with the explanation that Ben must have really hurt her.
* Effort justification: Liz reflects on her impromptu act of kindness where she prepares a romantic dinner for herself and Ben. She decides that the amount of effort she exerted must illustrate her deep love for Ben.
* New information: Liz comes across an article which claims that opposites do not last in the long term. Applying this principle to the opposite personalities possessed by herself and Ben, she decides to rationalise the belief rather than accept it by reassuring herself that they are soul mates and 'one in a million'.

'''Resulting cognitive dissonance'''

Liz may recognise aversive emotions such as tension and physical arousal.

'''Motivated behaviour to attain a consonant relationship'''

Liz rejects the call from Ben (avoidance) but attempts to resolve the conflict through a romantic dinner (approach).

'''Determinants of change'''

* Importance: Liz may reason that Ben is the biggest part of her life and this would influence her behaviour differently from if she regarded the relationship having run its course.
* Dissonance ratio: Liz may identify that her consonant reasons to stay in the relationship (‘we love each other’, ‘we have surpassed difficulties in the past’, ‘we have many fond memories’) outnumbers the dissonant reasons to dissolve the relationship (‘we have experienced a heated argument’, ‘we are both stubborn in fights’).
* Rationale: Liz may acknowledge that Ben’s recent occupation change and financial strains while he was locating a new position may have triggered or intensified the fighting.
* Reality: Liz may consider whether Ben really defines her as a person or whether she could benefit from putting more effort into excelling at her job or maintaining other friendships.
* Pain-cost ratio: Liz may justify withstanding the arguments with Ben and sleepless nights because she believes resolving the dissonance and continuing their relationship is worth the associated costs and pain.

'''Methods and outcomes of dissonance reduction or elimination'''

* Importance determinant: Ben is very important to me [[File:Arrow green2 en.svg|20px]] Remove dissonance by ceasing to argue and showing forgiveness through a hug.
* Rationale determinant: Ben's new occupation and the couple's financial strains are problematic [[File:Arrow green2 en.svg|20px]] Liz may reduce the importance of the dissonance by reasoning that situational factors are the underlying cause of the fight
* Dissonance Ratio determinant: We have more dissonance and reasons to end, rather than savor, the relationship [[File:Arrow green2 en.svg|20px]] Liz may add a consonant attitude by reasoning that the relationship does not define her as a person and she should be focusing on creating a balance in her life, for example, by exerting more effort in excelling at her job or maintaining her friendships.
* Reality determinant: Can working on the relationship and balancing other aspects of her life realistically work? [[File:Arrow green2 en.svg|20px]] Liz may increase the importance of the consonant attitude by scheduling in some relationship counselling and making times to see her friends
|}

Festinger's [[cognitive dissonance]] theory proposes that inconsistent attitudes, thoughts and behaviours about the self produce psychological discomfort (West &amp; Turner, 2007). As illustrated in the attribution theory, cognitive dissonance arises from an individual's desire to maintain consistency, control and predictability in their life. Dissonance is therefore aversive because it makes one's self-concept unstable which erodes the foundational base that people function from (Pemberton &amp; Sedikides, 2001). Accordingly, motivation arises from the need to harmonise the inconsistent relationship. The following section will outline the process of cognitive dissonance as depicted in Figure 2.

{{font|color=Purple|What causes self-concept change?}}

*'''Dissonance-arousing situations'''
Cognitive dissonance occurs when individuals appraise their attitudes, thoughts or behaviours as being incompatible, immoral or unreasonable. Reeve (2009) asserts that cognitive dissonance commonly arises under four circumstances; firstly, choice refers to the ultimatum individuals are faced with where they have to choose between two difficult options. Secondly, insufficient justification occurs when an individual has to explain an action which had little or no external prompting. Thirdly, effort justification defines a situation when an individual needs to reason why they exerted substantial effort for a particular behaviour. Fourthly, new information triggers cognitive dissonance because it can challenge or contradict one's beliefs.

The discomfort associated with cognitive dissonance incorporates both physiological and psychological arousal. Elkin and Leippe (1986) found that cognitive dissonance was a drive state with arousal reduction dependent upon a participant changing their attitude or becoming unaware of the dissonance after partaking in the task for a long duration of time. Similarly, Cooper, Zanna and Taves' (1978) study where participants completed a dissonance task after ingesting either a sedative or amphetamine. Results showed that sedatives attenuated the effects of dissonance in which an attitude change was no more prominent than the control group. Conversely, amphetamines augmented the effects of dissonance in which an attitude change was significantly more likely to occur.

Cognitive dissonance also comprises a psychological component of discomfort. Research indicates that dissonance triggers discrete, aversive feelings which are differentiated from positive or negative affect (Elliot &amp; Devine, 1994). The cause of the psychological discomfort relates back to the attribution theory, where individuals are uncomfortable with possessing a self-concept which is inconsistent and unpredictable. Thus, when an individual notices their attitude and behaviour are incongruent, it produces motivational properties to harmonise the inconsistency.

*'''Motivated behaviour to attain a consonant relationship'''

The key component behind the cognitive dissonance theory relates to its motivational properties (West &amp; Turner, 2007). When an individual experiences dissonance (providing it is of motivational magnitude), the individual is energized to enact either approach or avoidance behaviour. The particular direction to reduce dissonance will be discussed later, but it is important to note that at this point in the process an individual is energized to reduce the dissonance.

{{font|color=Purple|How is the self-concept changed/managed?}}

*'''Determinants of change'''

The determinants of approach or avoidance behaviour are governed by magnitudes of dissonance; the quantitative amount of dissonance an individual experiences including importance, dissonance ratio, rationale, reality and pain-cost ratio (Reeve, 2009; West &amp; Turner, 2007). Importance refers to how imperative the issue is to an individual; dissonance ratio defines the number of consonant attitudes relative to the dissonant attitudes; rationale refers to the reasoning exercised to explain the dissonance; reality describes an individual's appraisal of whether attitudinal or behavioural changes are realistic; and the pain-cost ratio simply refers to the amount of pain and costs required to eliminate a dissonance. Therefore, the determinants of change impact on whether an individual displays approach or avoidance behaviours to reduce or eliminate the dissonance.

*'''Methods and outcomes of dissonance reduction or elimination'''

The reduction or elimination of dissonance commonly occurs through the use of one of four methods including removal of dissonance, reduction of the importance of dissonance, addition of a consonant attitude or increase the importance of the new consonant attitude (West &amp; Turner, 2007).

[[File:Cognitive dissonance.JPG|thumb|center|600px|&lt;big&gt;Figure 2. Cognitive Dissonance Model&lt;/big&gt;]]

''{{font|color=Purple|Summary:}} one explanation for why people modify or change their self-concept is when they become aware of cognitive dissonance. As described by the attribution theory, people desire control, consistency and predictability over their attitudes, emotions and behaviour. Noticing an inconsistency between an attitude and behaviour therefore triggers physiological and psychological discomfort. Subsequently, people seek to harmonise this inconsistency through modifying or changing their attitudes or behaviours. When modification or change is successfully executed, individuals are able to clarify and consolidate particular self-schemas which make up the self-concept.''
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==={{font|color=Purple|Self-Discrepancy Theory}}===

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{{big2|'''Self-Discrepancy Theory Illustrated through Western Pop Music'''}}&lt;br&gt;
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{| border=1 cellspacing=0 cellpadding=5
|-
! [http://www.youtube.com/watch?v=lwDXdVNEmLc 'Empty'] by The Cranberries
|-
| ''&quot;All my plans fell through my hands,''

''They fell through my hands on me.''

''All my dreams it suddenly seems,''

''It suddenly seems,''

''Empty&quot;''
|-
|}
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* Illustrates an actual-ideal discrepancy. The persona expresses plans that they had forseen for themselves which end up dissolving. Thus, they have not achieved aspects of their ideal self-concept. Consequently, the persona expresses emptiness, a dejection-related emotion which reflects the self-discrepancy theory (Boldero et al., 2005)

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{| border=1 cellspacing=0 cellpadding=5
|-
! [http://www.youtube.com/watch?v=UnuTG5ygaow 'Don't let me get me'] by Pink
|-
| ''Tired of being compared to damn Britney Spears''

''She's so pretty, that just ain't me''

''&quot;Don't let me get me''

''I'm my own worst enemy''

''Its bad when you annoy yourself''

''So irritating''

''Don't wanna be my friend no more''

''I wanna be somebody else&quot;''
|-
|}
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* Illustrates an actual-ought discrepancy. The persona conveys the psychological discomfort experienced from believing they need to achieve the standards of celebrities, such as Britney Spears. Literature highlights the impact of the media and pressures to attain Western ideals including fame, achievement and wealth which consequently triggers psychological discomfort, particularly among youth (Dittmar, 2009; Johnson &amp; Krueger, 2006; Sanchez &amp; Crocker, 2005). In response to the persona's actual-ought discrepancy, they express feelings of irriation, which also reflects the self-discrepancy theory (McDaniel &amp; Grice, 2008).

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{| border=1 cellspacing=0 cellpadding=5
|-
! [http://www.youtube.com/watch?v=aY4Z2OFtl9U 'Perfect'] by Simple Plan
|-
| ''&quot;Hey Dad look at me''

''Think back and talk to me''

''Did I grow up according''

''To plan?''

''Do you think I’m wasting''

''My time doing things I''

''Wanna do?''

''But it hurts when you''

''Disapprove all along&quot;''
|-
|}
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* Illustrates an actual-ideal and actual-ought discrepancy. The persona communicates a perceived pressure to fulfil their father's expectations of what self-schemas they should possess (actual-ought). Furthermore, there is a discrepancy between what the father perceives as idyllic as compared to the persona (actual-ideal). Consequently, the persona conveys a mixture of frustration (agitation-related emotions) and sadness (dejection-related emotions) which reflects the self-discrepancy theory (Boldero et al., 2005).
|}

The Self-Discrepancy Theory addresses the motivational and emotional properties triggered from actual, ideal and ought self discrepancies (Crisp &amp; Turner, 2010). The actual self refers to an individual’s current self-concept, the ideal self to an individual’s idyllic self-concept and the ought self to what an individual perceives their self-concept should encompass due to obligation or responsibility (McDaniel &amp; Grice, 2008). Discrepancies consequently arise when the selves conflict which triggers the psychological discomfort outlined in the attribution theory due to the inconsistency and instability of one’s self-concept. Discrepancies stimulate negative emotional reactions including dejected-related emotions in response to actual-ideal inconsistencies and agitation-related emotions in response to actual-ought inconsistencies (Boldero, Moretti, Bell &amp; Francis, 2005). Self-discrepancies have also been proposed as an indirect contributor to suicidal ideation, mood disorders and [[anxiety]] disorders (Cornette, Strauman, Abramson &amp; Busch, 2009; McDaniel &amp; Grice, 2008).

The self-discrepancy theory demonstrates how actual-ideal-ought discrepancies can impact on one's self-concept and emotional displays. The theory also provides insight into how these discrepancies motivate behaviour. Specifically, discrepancies can motivate us to change or modify our self-schemas to bridge the gap between our actual-ideal or actual-ought selves. For example, a shy person (actual self) wanting to be confident (ideal self) may enrol in an assertiveness course, practise saying no or gradually become comfortable with delivering speeches through relaxation techniques. The discrepancy has therefore motivated them to engage in strategising and productive behaviours.

The ability for self-discrepancies to motivate behaviour is evident in a diverse range of real life contexts. Wilson, Mack and Grattan (2008) propose that discrepancies between actual-ideal and actual-ought selves can motivate physically unfit or overweight individuals to engage in exercise and diet schedules. Similarly, the theory can be applied to individuals who engage in excessive alcohol consumption and subseqently improve drinking behaviour through a greater self-awareness of actual-ideal standards (McNally, Palfai &amp; Kahler, 2005).

Pentina, Taylor and Voelker (2009) also found that self-discrepancies play a role in young females' decisions to undergo cosmetic surgery. Specifically, actual-ideal discrepancies motivated individuals to seek cosmetic surgery possibly because they had internalised body image values which indicated that their physical self-schema was inadequate. However, actual-ought discrepancies actually reduced cosmetic surgery-seeking behaviours possibly because it leads people to question and avoid risky and commercialised consumption choices. Importantly, family support tended to reduce cosmetic surgery-seeking behaviour while peer support increased it which suggests that additional factors contribute to motivated behaviour. Thus, self-discrepancies play a key role in energising and directing behaviour, however, contemporary research suggests this is moderated by individual factors.

The type and intensity of motivated behaviour produced by self-discrepancies appears to be moderated by individual characteristics (e.g. Lodewyk, Gammage &amp; Sullivan, 2009; Roney &amp; Sorrentino, 1995). Roney and Sorrentino (1995) firstly outline numerous biological, cognitive and psychological factors which implicate motivated behaviour. Biological predispositions such as personality tendencies can direct behaviour in two ways; individuals can possess a tendency to simply avoid discrepancies rather than engage in self-regulation to monitor and improve actual-ideal-ought discrepancies, and individuals often innately exhibit an uncertainty-oriented or certainty-oriented tendency. Uncertainty-oriented individuals are energised to seek out new information about themselves in a process called self-assessment, which subsequently enables them to engage and thrive on amending self-discrepancies.

Conversely, certainty-oriented individuals are energised to maintain consistent self-schemas and subsequently avoid self-assessment to prevent the confrontation of conflicting information about themselves (Deckers, 2004). Thus, uncertainty-oriented individuals often embrace self-discrepancies while certainty-oriented individuals avoid self-discrepancies which provides one explanation for, say, those individuals who devise and maintain a weight loss program and those who do not.

Roney and Sorrentino (1995) also outline several cognitive factors which influence motivated behaviour. Firstly, their study demonstrated that regardless of certainty or uncertainty orientation, individuals who possessed a success-oriented mindset performed better academically than individuals who possessed a failure-threatened mindset. Thus, when individuals set out to reduce a self-discrepancy, approaching the situation with a positive and success-oriented mindset is paramount in effective performance and goal attainment.

A second cognitive factor explored by Roney and Sorrentino involved the process of priming individuals with ideal or ought selves prior to the academic task. Results showed improved performance when individuals were primed with ideal selves and poorer performance when primed with ought selves. This supports Pentina et al. (2009) whose findings suggest that ideal selves stem from an individual's internalised, private beliefs while ought selves stem from an individual's perception of socially desirable beliefs and behaviours.

Roney and Sorrentino also posit that psychological factors can influence motivated behaviour. Their findings suggested that harder goals do not necessarily lead to superior performance as advocated by the goal-setting theory (Locke &amp; Latham, 2006). This is because difficult goals can lead to greater self-discrepancies over time in addition to intensified negative emotions. Thus, self-discrepancies can motivate productive behaviour on the proviso that goal difficulty is reasonable and attainable.

A final dimension which implicates motivated behaviour involves social influences. Lodewyk et al. (2009) emphasise the effects of Western ideals such as achievement and physical attractiveness pressures. Specifically, extreme self-discrepancies can lead to unconstructive, rather than productive, behaviours where the actual-ideal-ought selves trigger anxiety, low self-efficacy and poor outcomes. Similarly, Pentina et al. (2009) highlight the influence of social support on motivated behaviour as mentioned previously. The relationship between self-discrepancies and motivated behaviour can therefore trigger productive or counter-productive behaviour. The direction of behaviour appears to be moderated by biological, cognitive, psychological and social factors in combination with the time-frame, type and difficulty of the self-discrepancy.

{{Robelbox|theme={{{theme|3}}}|title=Self-Discrepancy Theory: In Focus}}
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Body Dysmorphic Disorder is a somatoform disorder which is characterised by a preoccupation with an imagined or slight physical defect, such as the size or shape of the nose (Buhlmann &amp; Wilhelm, 2004). The disorder leads to impaired functioning and distress and is commonly comorbid with other disorders such as major depression, social phobia or anxiety (Durand &amp; Barlow, 2010). Body dysmorphic disorder has been associated with the self-discrepancy theory as research suggests sufferers demonstrate inconsistencies between their actual-ideal and actual-ought selves (e.g. Veale, Kinderman, Riley &amp; Lambrou, 2003).

Specifically, one participant recorded on a 10-point scale that their breasts were too saggy (10-points; actual self), they their ideal breasts would be firm, high and with small nipples (10-points; ideal self), and their ought breasts should also be firm, high and with small nipples (10-points; ought self). Thus, body dysmorphic disorder is intimately related and intensified by discrepancies between what a sufferer perceives they should, ideally and actually look like. Sufferers recognise inconsistencies within their self-concept, stimulated by dejected or agitated emotions. Consequently, a preoccupation with ‘fixing’, checking or avoidance of the defect is evident as sufferers attempt to cope with the discrepancy.
  &lt;/div&gt;
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==={{font|color=Purple|Possible Selves}}===

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{{big2|'''Possible Selves and KidZania'''}}
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Possible Selves and [http://www.kidzania.com/ KidZania]: The development of possible selves is evident from childhood with contemporary society further promoting this process with the establishment of facilities such as KidZania. KidZania was founded in Mexico, 1999, and is now located in 14 other cities. The facility is designed to provide children aged 3+ with work experience, independence, social skill development and an understanding of sensible money use. Children come to the facility with their parents and participate in one of the 90 'jobs' for an entire work shift (9am-3pm or 4pm-9pm). They receive training and specific tasks related to their job, such as preparing and cooking a meal as a chef, and on completion of the shift are paid in KidZos which they can use to purchase products and services in the gift shop. Thus, it enables children to 'test out' their possible selves and gain a clearer understanding of the ideal selves they wish to strive for.
|}

Like self-discrepancies, possible selves energise and direct behaviour. Possible selves refer to the totality of self-schemas one does and does not wish to possess to form a coherent and consistent self-concept (Deckers, 2004). For example, an individual may wish to be an excellent conversationalist but not desire to have superior physical fitness. Possible selves subsequently direct an individual's attention to exert effort and persistence in strategically planning and monitoring goal attainment (Reeve, 2009). For instance, the individual in the former example would exert effort in improving their language skills, humour and ability to be personable.

Research suggests that individuals are more likely to think about future rather than past selves (Erikson, 2007), consider positive rather than negative selves (vanDellen &amp; Hoyle, 2008) and believe they will become their positive rather than negative selves (Hart, Fegley &amp; Brengelman, 1993). However, these tendencies are moderated by an individual's level of optimism and pessimism, with optimism being positively correlated to these tendencies and pessimism being negatively correlated (Deckers, 2004). Failure to substantiate one's possible self commonly results in one of two outcomes; the individual rejects the possible self or develops strategies and goals to achieve the posisble self (Reeve, 2009). Thus, the role of possible selves is to enable an individual to evaluate their current self and strive for a future self of more value.

==={{font|color=Purple|Self-Esteem}}===
[[File:Suryanamaskar.gif|left|thumb|150px|&lt;big&gt;The ideal self..&lt;/big&gt;]]

[[File:Woman leaning against car in Linz, Austria.jpg|left|thumb|150px|&lt;big&gt;..and actual self&lt;/big&gt;]]

[[Self-esteem]] is a final factor which contributes to an individual's possible selves and the direction of their motivated behaviour. Self-esteem refers to the positive or negative evaluation of oneself based on how well the current self fares against the possible selves (Deckers, 2004). Specifically, self-esteem has been expressed through the following equation (Rodriguez, Wigfield &amp; Eccles, 2003):

The equation defines pretensions as an individual's imagined possible selves and success as achievements towards becoming these possible selves. For example, Jenny may aspire to obtaining a 5.00 GPA which constitutes her possible self, and a success would involve achieving the 5.00 GPA. The theory subsequently proposes that increasing one's self-esteem requires either the reduction of pretensions or the increase of successes. Conversely, a decline in self-esteem is claimed to involve either an increase in pretensions or a decrease in successes (Rodriguez et al., 2003). Importantly, self-esteem is relative to the extent that an individual attributes importance to the possible self. If Jenny attributed no importance to academic schemas, then a failure to attain a 5.00 GPA would not necessarily result in reduced self-esteem (Deckers, 2004). Self-esteem therefore influences the direction and intensity of motivated behaviour and the subsequent selection and development of self-schemas which comprise one's self-concept.

Self-esteem influences motivated behaviour in different ways depending on whether one's self-esteem is positive or negative (Elliott, 2007). Cross and Markus (1994) illustrated this concept through a study which examined task performance in students who were schematic and aschematic in problem-solving. The results demonstrated that an individual with a positive self-esteem (schematic) possesses self-schemas which facilitate encoding, evaluation and retrieval of domain-relevant information. Schematic individuals were able to quickly reject inconsistent feedback (e.g. unbothered by an incorrect answer), were primed to respond in a particular way (e.g. quick and confident judgements, high information-processing) and were more sensitive and attentive to domain-relevant information (e.g. responded productively to feedback, utilised problem-solving skills). Thus, schematic individuals in a particular schema domain possess a superior ability to form and attain possible selves in that domain. Furthermore, motivated behaviour is action-oriented which is complemented by high self-efficacy.

Conversely, individuals with a negative self-esteem (aschematic) produced comparative results in the first task (which illustrates equal performance) but showed increasingly poorer performance throughout the remainder of the experiment. This is because they appraised the task as beyond their competence level which triggers avoidance reactions including reduced concentration, a lack of enjoyment and less exerted effort. Interestingly, aschematic participants who were given negative feedback throughout the task often improved their performance. Cross and Markus propose this result reflects an individual's fear that a problem-solving schema is indicative of general intelligence. When participants were consciously reminded of their performance, they were more attentive and exerted more effort than if they were not given feedback.

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{{big2|'''Self-Discrepancy Theory and Sun-Tanning'''}}
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One practical application of the self-discrepancy theory designed to improve human well-being involves the reduction of sun-tanning. Shoveller, Lovato, Young and Moffat's (2003) research indicates that making intentional sun-tanners consciously aware of their actual, ideal and ought selves can firstly increase self-awareness. Secondly, by providing sun-safe education in addition to challenging Western ideals of beauty, participants demonstrated significant reductions in sun-tanning.
[[File:Sun tanning lying on her stomach.jpg|centre|200px]]
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The importance of self-esteem can be summarised in the following example:

[[File:Arrow r.svg|15px]] An individual's self-schema in a particular domain arouses perceived abilities and skills. For example, a social skill domain for Tim arouses his perception that he is personable, funny and competent in this domain.

[[File:Arrow r.svg|15px]] Appraisal of the self-schema triggers the development and activation of possible selves. For example, Tim develops possible selves, such as being popular, and this self is activated when he is in a social situation.

[[File:Arrow r.svg|15px]] The development and activation of positive possible selves subsequently primes task-relevant thoughts, emotions and actions. For example, Tim becomes attentive to people's body language when in a social situation and channels in on skills such as being warm, empathetic and humorous.

[[File:Arrow r.svg|15px]] The activation of task-relevant skills enables an individual to utilise their skills in conjunction with increased self-efficacy which enables effective performance and self-concept consolidation. For example, Tim tunes into necessary social skills and is therefore able to utilise his skills, exert effort and persistence to attain his goal and in turn reaffirm his self-concept.

''{{font|color=Purple|Summary:}} the self-discrepancy theory provides a second explanation for why and how individuals change their self-concept. Like cognitive dissonance theory, it explains self-concept change through self appraisals which produce motivational and emotional properties due to an inconsistent and unpredictable self. When an individual becomes aware of a discrepancy they cognitively appraise it with agitated-related emotions (actual-ought discrepancies) or dejected-related emotions (actual-ideal discrepancies). It is the cognitive appraisal and subsequent aversive emotions that produce the motivation behind goal setting and strategy formulation to modify or change one's self-schema or general self-concept. However if one's ideal or ought self is too far removed from their actual self, counter-productive behaviour can occur. Similarly, possible selves generate motivational and emotional properties in the form of approach or avoidance behaviour. Self-esteem tends to moderate the motivational and emotional properties of self-discrepancies and possible selves. Specifically, a positive self-esteem can trigger task-relevant thoughts, self-efficacy, optimisation of individual skills and abilities and self-concept growth. Conversely, a negative self-esteem can hinder self-concept development and stunt growth in self-schema domains.''
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==={{font|color=Purple|Self-Evaluation Maintenance Model}}===
[[File:Social Reflection.JPG|right|thumb|350px|&lt;big&gt;Figure 1.1. Factors Contributing to Social Reflection&lt;/big&gt;]]

The Self-Evaluation Maintenance Model posits that challenges to one's self-concept and self-esteem motivates the engagement in either social comparison or social reflection depending on how the individual appraises the situation (Crisp &amp; Turner, 2010). Social comparison involves the process of comparing oneself to another and subsequently using that analysis to define and evaluate their/your self-schemas (Pemberton &amp; Sedikides, 2001). Comparisons often involve either upward comparisons where a person compares themselves to someone perceived as superior to them or downward comparisons where a person compares themselves to someone they perceive as inferior to them (Crisp &amp; Turner, 2010). For example, social comparison is common amongst close work colleagues as it provides explicit performance judgements in the form of work evaluations, promotions or bonuses (Tesser, Millar &amp; Moore, 1988).

Alternatively, social reflection refers to the bolstering of one’s self-esteem and development of the self-concept through internalising the achievements of others (Pemberton &amp; Sedikides, 2001). For example, parents often engage in social reflection where they ‘bask in reflected glory’ from the achievements of their child such as sporting or academic accomplishments (Hannawa &amp; Spitzberg, 2009).

[[File:Social Comparison.JPG|right|thumb|350px|&lt;big&gt;Figure 1.2. Factors Contributing to Social Comparison&lt;/big&gt;]]

Two key factors are argued to determine whether social comparison or social reflection is utilised, as detailed in Figures 1.1 and 1.2. However, recent research suggests performance, closeness and information are additional determinants (Beach et al., 1996; Pemberton &amp; Sedikides, 2001). Specifically, individuals are more likely to engage in social comparison if they perceive their past or future performance is threatened by others and/or if they are provided with performance information about the other person which suggests competence (Pemberton &amp; Sedikides, 2001). Similarly, individuals are more likely to engage in social comparison if they possess a close relationship to the other person, as opposed to being strangers (Beach et al., 1996). Some research goes further by arguing that all other determinants of behaviour are irrelevant if the comparison is with a stranger (e.g. Tesser et al., 1988). This is because strangers do not directly implicate an individual’s self-concept or self-esteem as they are separate from one’s micro world and therefore hold no value in relation to one’s social environment.

* Note: Self-concept refers to the mental representations of oneself while self-esteem defines the subjective negative or positive appraisal of oneself (Crisp &amp; Turner, 2010).

The Self-Evaluation Maintenance Model subsequently posits that social reflection or comparison triggers motivated behaviour (Deckers, 2004). Social reflection motivates the person to engage in positive thoughts about him/herself (they are a part of the success), energises positive emotions (such as joy and pride in the achievement) and directs positive behaviour (such as celebratory actions). Alternatively, social comparison motivates four main defense strategies which aim to protect and maintain a positive self-concept, as displayed in Table 2(Crisp &amp; Turner, 2010).

{{center top}}'''Table 2 Strategies to Combat the Effects of Social Comparison'''{{center bottom}}
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{| border=1 cellspacing=0 cellpadding=5
|-
! Strategy
! Example
|-
| 1. Exaggerate the abilities of the other person
| &quot;My work colleague, Sue, is beyond intelligent - She went to Harvard University so it's unrealistic to compare myself to her&quot;
|-
| 2. Distance yourself from the other person
| &quot;I don't particularly agree with Sue's decisions anyway so I'm applying for a department transfer&quot;
|-
| 3. Devalue the importance of the domain
| &quot;Sue is far too work-orientated - At least I have a superior level of fitness and a busy social life&quot;
|-
| 4. Compare yourself to a different person	
| &quot;I am more competent at my job than my other co-workers&quot;
|-
|}
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''{{font|color=Purple|Summary:}} this model provides another explanation for why and how people modify/change their self-concepts. However, rather than self comparisons it addresses social comparisons and how an individual can maintain a stable self-concept and thus psychological well-being. Specifically, perceptions of oneself compared to others can trigger social comparison where an individual bolsters their self-concept through downward comparison or threaten the consistency and stability of their self-concept through upward comparison. Alternatively, individuals can engage in social reflection which acts as a protective and self-enhancing technique.''
{{Only in print|'''Snow White and the Seven Dwarfs: An illustrative example of Social Comparison'''}}
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[[File:José Ángel Santana.jpg|left|thumb|200px|&lt;big&gt;Possibly not the most accurate representation of the fairest one of all!&lt;/big&gt;]]

The well-known 1937 fairytale provides an illustrative example of social comparison through two characters; The Queen and the Magic Mirror. In an attempt to maintain self-verification and self-enhancement, The Queen frequently asks the Magic Mirror (which can speak nothing but the truth):
[[File:Magic mirror.jpg|right|thumb|150px|&lt;big&gt;The Magic Mirror&lt;/big&gt;]]

''Mirror, Mirror on the wall,''
''Who is the fairest one of all?''

''You, O Queen, are the fairest one of all''

Thus, The Queen is able to maintain a stable and consistent self-concept through feedback from the Magic Mirror. However, as the tale unfolds, The Queen asks once again and is subsequently denied the fairest one of all as depicted in this [http://www.youtube.com/watch?v=g_Fbynm6M-4&amp;feature=related clip]

Consequently, The Queen engages in social comparison as she perceives Snow White as a threat to her relevant domain (fairest one of all) and becomes uncertain of her abilities (thus formulates an evil plan). Furthermore, The Queen perceives Snow White as a threat to her future performance and status, holds a somewhat close relationship with Snow White (as she is her maid) and possesses explicit information that Snow White is ‘fairer’ than her (Dirks, 2010).
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== Chapter Recap ==

* Self-concept permeates all facets of an individual's life and is a primary contributor to psychological well-being. Motivation is instrumental in energising and directing an individual to work out who they are (self-assessment), create a consistent and accurate hierarchy of self-schemas (self-verification) and enable control and psychological well-being (self-enhancement). The purpose of the self-concept therefore reflects the human tendency to seek consistency, control and predictability over their attitudes, emotions and behaviours.
* Self-concept develops from a range of elements including biological, brain structure, neurotransmitter, socio-cultural and psychological components. Each individual comprises a particular combination of these elements which subsequently produces an unique self-concept and motivated attitudes, emotions and behaviours.
* Self-concept incorporates a core part and ever-changing outer part which interact to produce motivated attitudes, emotions and behaviours. Change is often necessary for self-concept development and stability as it enables an individual to assess, modify and reaffirm their self-schemas. However, changes tend to be slight, gradual differences which enable the core to maintain effective functioning.
* Self-determination theory proposes that humans possess an innate tendency to strive for self-concordant goals. This enables the fulfilment of psychological needs, such as striving for autonomy through life goals, competence through environmental mastery and relatedness through positive interpersonal relations. Together, self-concordant goals and psychological satisfaction encourage the development of intrinsic motivation which enables persistence, creativity, conceptual understanding and optimal functioning. This subsequently places an individual in a position to attain facets of psychological well-being including self-acceptance, positive interpersonal relations, autonomy, environmental mastery, purpose in life and personal growth.
* The cognitive dissonance theory asserts that an inconsistent attitude and behaviour produces physiological and psychological discomfort which can trigger self-concept modification or change. When attitude or behaviour modification or change is successfully executed, individuals are able to clarify and consolidate particular self-schemas which make up the self-concept.
* The self-discrepancy theory outlines self-concept change and management from personal reflections on inconsistent actual-ideal and actual-ought selves. When an individual becomes aware of a discrepancy they cognitively appraise it with agitated-related emotions (actual-ought discrepancies) or dejected-related emotions (actual-ideal discrepancies). It is the cognitive appraisal and subsequent aversive emotions that produce the motivation behind goal setting and strategy formulation to modify or change one's self-schema or general self-concept. However if one's ideal or ought self is too far removed from their actual self, counter-productive behaviour can occur.
* Possible selves generate motivational and emotional properties in the form of approach or avoidance behaviour. Self-esteem tends to moderate the motivational and emotional properties of self-discrepancies and possible selves. Specifically, a positive self-esteem can trigger task-relevant thoughts, self-efficacy, optimisation of individual skills and abilities and self-concept growth. Conversely, a negative self-esteem can hinder self-concept development and stunt growth in self-schema domains.
* The self-evaluation maintenance model highlights how social feedback can trigger self-concept change and management. Perceptions of oneself compared to others can trigger social comparison where an individual bolsters their self-concept through downward comparison or threatens the consistency and stability of their self-concept through upward comparison. Alternatively, individuals can engage in social reflection which acts as a protective and self-enhancing technique.
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==Test Yourself: Answering the Chapter Focus Questions==
&lt;quiz&gt;
Self-concept describes:
|type=&quot;()&quot;}
- One's social roles
+ Mental representations of oneself
- Who you would like to become

{Motivation describes:
|type=&quot;()&quot;}
- The energy behind behaviour
- The direction behind behaviour
+ All of the above

{Motivation energises and directs an individual to work out who they are (self-...........), create a consistent and accurate hierarchy of self-schemas (self-.............) and enable control and psychological well-being (self-...........):
|type=&quot;()&quot;}
- Self-enhancement, self-verification, self-assessment
+ Self-assessment, self-verification, self-enhancement
- Self-assessment, self-enhancement, self-verification

{An individual's self-concept permeates all facets of his/her life and is a key motivational factor underlying attitudes, emotions and behaviours in the hope to attain:
|type=&quot;()&quot;}
+ Psychological well-being
- Eudaimonic well-being
- Hedonic well-being

{Personality traits, self-recognition and peer groups refer to which developmental elements (in order):
|type=&quot;()&quot;}
- Psychological, biological, socio-cultural
- Biological, socio-cultural, psychological
+ Biological, psychological, socio-cultural

{Which of the following statements are false:
|type=&quot;()&quot;}
+ Self-concept change often occurs in all self-schema domains at once
- Self-concepts often comprise a core stable part and an ever-changing outer part
- Self-concept change is often slight and gradual

{The cognitive dissonance theory asserts that individuals become aware of undesirable self-concepts when:
|type=&quot;()&quot;}
- A stressful life event is encountered
+ An attitude is inconsistent with a behaviour
- An individual enters a new life stage

{Actual-ideal discrepancies often trigger which type(s) of emotions:
|type=&quot;()&quot;}
+ Dejected-related emotions
- Agitated-related emotions
- All of the above

{Which of the following statements are false:
|type=&quot;()&quot;}
- Negative self-esteem can lower performance
- Positive self-esteem increases the probability of forming and attaining possible selves
+ Poor performance in a domain irrelevant self-schema produces negative self-esteem

{Social reflection most often occurs when:
|type=&quot;()&quot;}
+ The domain is irrelevant and abilities are certain
- The domain is irrelevant and abilities uncertain
- The domain is relevant and abilities certain
&lt;/quiz&gt;
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==See also==
* [[Motivation and emotion/Book/2010/Motivation and goal setting|Motivation and goal setting]] (Book chapter, 2010)
* [[Motivation_and_emotion/Textbook/Motivation/Self-actualisation|Self-actualisation]] (Textbook chapter)
* [[Motivation_and_emotion/Textbook/Motivation/Achievement_motivation|Achievement motivation]] (Textbook chapter)
* [[Motivation_and_emotion/Textbook/Emotion/Emotional_stability|Emotional stability]] (Textbook chapter)
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Salmivalli, C. (1998). Intelligent, attractive, well-behaving, unhappy: The structure of adolescents self-concept and its relations to their social behavior. Journal of Research on Adolescence, 8, 333-354. doi: 10.1207/515327795jra0803-3

Sanchez, D. T., &amp; Crocker, J. (2005). How investment in gender ideals affects well-being: The role of external contingencies of self-worth. Psychology of Women Quarterly, 29, 63-77. doi: 10.1111/j.1471-6402.2005.00169.x

Shoveller, J. A., Lovato, C. Y., Young, R. A., &amp; Moffat, B. (2003). Exploring the development of sun-tanning behavior: A grounded theory study of adolescents' decision-making experiences with becoming a sun tanner. International Journal of Behavioral Medicine, 10, 299-314. doi: 10.1207/S15327558IJBM1004_2

Showers, C. J., Abramson, L. Y., &amp; Hogan, M. E. (1998). The dynamic self: How the content and structure of the self-concept change with modd. ''Journal of Personality and Social Psychology, 75,'' 478-493. doi: 10.1037/0022-3514.75.2.478

Tenenbaum, H. R., &amp; Leaper, C. (2002). Are parents' gender schemas related to their children's gender-related cognitions? A meta-analysis. Developmental Psychology, 38, 615-630. doi:10.1037/0012-1649.38.4.615

Tesser, A., Millar, M., &amp; Moore, J. (1988). Some affective consequences of social comparison and reflection processes: The pain and pleasure of being close. Journal of Personality and Social Psychology, 54, 49-61. doi: 10.1037/0022-3514.54.1.49

Thrash, T. M., &amp; Elliot, A. J. (2002). Implicit and self-attributed achievement motives: Concordance and predictive validity. ''Journal of Personality, 70,'' 729-755. doi: 10.1111/1467-6494.05022

Veale, D., Kinderman, P., Riley, S., &amp; Lambrou, C. (2003). Self-discrepancy in body dysmorphic disorder. British Journal of Clinical Psychology, 42, 157-169. doi: 10.1348/014466503321903571

vanDellen, M. R., &amp; Hoyle, R. H. (2008). Possible selves as behaioral standards in self-regulation. Self and Identity, 7, 295-304. doi: 10.1080/15298860701641108

van der Zee, K., Thijs, M., &amp; Schakel, L. (2002). The relationship of emotional intelligence with academic intelligence and the big five. European Journal of Personality, 16, 103-125. doi: 10.1002/per.434

von Wyl, A., Perren, S., Braune-Krickan, K., Simoni, H., Stadlmayr, W., Burgin, D., &amp; von Klitzing, K. (2008). How early triadic family processes predict children’s strengths and difficulties at age three. European Journal of Developmental Psychology, 5, 466-491. doi: 10.1080/17405620600989701

West, R., &amp; Turner, L. H. (2007). Introduction to communication theory: Analysis and application (3rd ed.). New York: The McGraw-Hill Companies.

Wilson, P. M., Mack, D. E., &amp; Grattan, K. P. (2008). Understanding motivation for exercise: A self-determination theory perspective. Canadian Psychology, 49, 250-256. doi: 0.1037/a0012762
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      <text bytes="7548" sha1="jcvscx1pl0wkjwjo8v96lq792eqpjud" xml:space="preserve">==Feedback==
# Overall this is a promising draft so far, with lots of detailed and interesting information. The main areas to improve are to reduce the overall length, separate the examples from the main body, shorten overly long paragraphs. The introduction and conclusion need to work on establishing the key foci for the chapter. Make it clear in each section how it addresses the chapter's stated aims.
# I started separating the Liz and Ben example from the body text - it is a really good ideas to have a rich example like this, but I think it will work better if you run the example in separate side-boxes. I suggest do something similar throughout - makes it easier for a reader to digest.
# Avoid overly long paragraphs - generally, 3 to 5 sentences is a good paragraph length.
# Self-concept motivates emotions - is this heading accurate? (I think it could better reflect the content) (you only need to relate self-concept to motivation - not emotions. So, maybe simplify.)
## is self-concept seen here as motivating emotions - or just causing them? Do the cognitions causes emotions then cause motivation? Try to explain a bit more how this section connects with the overall topic (are there focus questions to be answered?). Or perhaps this is a bit of a side detour here?
## I think this might work better if you have the pop songs and related commentary in a separate side box and then have a short section which focuses on relevant theory and research (with citations).
# Self-Discrepancy Theory: In Focus - I'd suggest running this in a sidebox too, running down the page with the theory/research content next to it (as for the previous sections).
# Self-Discrepancy + Possible selves = almost 2000 words! (Self-esteem is another 700 words). I didn't read all of this - I get the sense that it could be abbreviated. What is the focus question you are trying to answer? Explain this question first and why it's important - then provide an answer drawing on and explaining relevant theory and research, with an example in a sidebox.
# Self Determination Theory could be relevant in many places here - e.g., the need for competence relates to self-discrepancy and possible selves
# Self-Evaluation Maintenance Model - there was no previous announcement of this model or particularly flow/connection. As you redraft make sure to set up in your introduction what the key questions and topics are going to be and why the chapter focuses on these ones in particular.
# References
## May be of interest: Showers, C. J., Abramson, L. Y., &amp; Hogan, M. E. (1998). The dynamic self: How the content and structure of the self-concept change with mood. Journal of Personality and Social Psychology, 75, 478-493. 
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:14, 1 November 2010 (UTC)

==Feedback 2==
Well done on advancing this draft. It is already a significant and substantial work. I've looked quickly through it again - my rough sense is this is DI-quality with potential for higher. Feedback below:
# Introduction
## Shaping up nicely, I gave it a tweak.
## Explain or link to info about those two types of well-being. 
## Check the link at the end.
## Add an overview of the focus questions the chapter seeks to address
# Boxes - I like these; simple, clear interesting. I'd probably increase their widths a bit (it will look different on different screens - here they are a bit narrowly squished).
# Definitions - Rich and useful; well done
# How Does the Self-Concept Develop? onwards - the rest of this content is substantial, interesting, and well referenced with clear focus on theory. The main areas to work on are
## flow and integration between the sections (e.g., by explaining the focus questions in the introduction and the intro/conclusion to each section.
## My other main suggestion is to as much as possible keep drawing the connection and relevant to motivation (in all sections).
# Where possible, keep the overall length shorter - not a big deal and I know this is hard. But for one other chapter I've suggested putting some related but supplementary material into subpages (but their draft was much longer).
# Since the chapter is already quite substantial I'd be inclined to have relatively short sections for &quot;Do Self-Concepts Vary Across Cultures?&quot; and &quot;How Stable is the Self-Concept and Why is Stability Important?&quot; - these are important topics for self-concept, but probably not as important as How does SC develop and how can it be changed for this chapter. If some whole sections were to be left out (an option) then it would be these two sections - or they could be treat them as supplementary topics.
# The pop-songs are useful - leave them in - but I would just separate them out of the main text body into a separate box (as you've done)
# I think there is some literature about different types of self-discrepancy, although a lot focuses on actual-ideal. I'm not how different actual-ought discrepancies would be from actual-ideal discrepancies - they sound quite similar to me.
# Keep the self-maintenance model but yes, make it clear in the intro/conclusion how it connects in with the focus questions and rest of the content - and draw connection with motivation.
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:51, 5 November 2010 (UTC)

{{MECF|10|
#Congratulations. This is a very impressive chapter. It is long, but it has been very well developed, from plan to draft to final draft with feedback sought and addressed along the way. Importantly the chapter does an excellent job covering relevant theory and research in an indepth and well-illustrated way. In so doing, the chapter addresses the underlying question about the relation between self-concept and motivation.
#I've made only very minor edits to improve the chapter.|10|
#A wide array of relevant theories were considered and impressively, there was a concerted effort to weave these theories together where appropriate and summarise them in the chapter recap.
#comment|10|
#Whilst there in an abundance of self-concept research, appropriate studies were selected and used to illustrate key points. The chapter is well cited and referenced.
#comment|10|
#The chapter was very well written and the layout with tables, figures, images, features boxes, quiz etc. further enhanced the communication and interactivity of the content.
#Typographical, grammatical and spelling errors were almost entirely absent.
# Some paragraphs were overly long (e.g., 200+ words).
#APA style was appropriately followed - I only fixed Table and Figure numbers, plus italics should be used in the references.
#comment}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:12, 23 November 2010 (UTC)

&lt;!-- Official multimedia presentation feedback --&gt;
{{MEMF|6|
#The script was well developed - but was too much for a 5 min. presentation - it was very dense and too fast.
#The audio volume was barely audible.
#Images were used effectively.
#Slow down, leave longer pauses between sentences. (This would probably require being more selective of content to focus on key points)
#Use more tonal variation (it was rather monotone)
#Well done on using so many free to use images, but image sources weren't acknowledged.
|7|
|6|
|5|
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:24, 13 December 2010 (UTC)</text>
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== References ==
&lt;/noinclude&gt;
&lt;div style=&quot;padding-left: 2em; text-indent: -2em&quot;&gt;
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Pentina, I., Taylor, D. G., &amp; Voelker, T. A. (2009). The roles of self-discrepancy and social support in young females' decisions to undergo cosmetic procedures. ''Journal of Consumer Behaviour, 8,'' 149-165. doi:10.1002/cb.279

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Rodriguez, D., Wigfield, A., &amp; Eccles, J. (2003). Changing competence perceptions, changing values: Implication for youth sport. ''Journal of Applied Sport Psychology, 15,'' 67-81. doi: 10.1080/10413200305403

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Rudman, L. A., &amp; Phelan, J. E. (2010). The effect of priming gender roles on women's implicit gender beliefs and career aspirations. ''Social Psychology, 41,'' 192-202. doi: 10.1027/1864-9335/a000027

Ryan, R. M., &amp; Deci, E. L. (2000). The darker and brighter sides of human existence: Basic psychological needs as a unifying concept. ''Psychological Inquiry, 11,'' 319-338. doi: 10.1207/S15327965PLI1104_03

Salmivalli, C. (1998). Intelligent, attractive, well-behaving, unhappy: The structure of adolescents self-concept and its relations to their social behavior. ''Journal of Research on Adolescence, 8,'' 333-354. doi: 10.1207/515327795jra0803-3

Sanchez, D. T., &amp; Crocker, J. (2005). How investment in gender ideals affects well-being: The role of external contingencies of self-worth. ''Psychology of Women Quarterly, 29,'' 63-77. doi: 10.1111/j.1471-6402.2005.00169.x

Shoveller, J. A., Lovato, C. Y., Young, R. A., &amp; Moffat, B. (2003). Exploring the development of sun-tanning behavior: A grounded theory study of adolescents' decision-making experiences with becoming a sun tanner. ''International Journal of Behavioral Medicine, 10,'' 299-314. doi: 10.1207/S15327558IJBM1004_2

Tenenbaum, H. R., &amp; Leaper, C. (2002). Are parents' gender schemas related to their children's gender-related cognitions? A meta-analysis. ''Developmental Psychology, 38,'' 615-630. doi:10.1037/0012-1649.38.4.615

Tesser, A., Millar, M., &amp; Moore, J. (1988). Some affective consequences of social comparison and reflection processes: The pain and pleasure of being close. ''Journal of Personality and Social Psychology, 54,'' 49-61. doi: 10.1037/0022-3514.54.1.49

Veale, D., Kinderman, P., Riley, S., &amp; Lambrou, C. (2003). Self-discrepancy in body dysmorphic disorder. ''British Journal of Clinical Psychology, 42,'' 157-169. doi: 10.1348/014466503321903571

vanDellen, M. R., &amp; Hoyle, R. H. (2008). Possible selves as behaioral standards in self-regulation. ''Self and Identity, 7,'' 295-304. doi: 10.1080/15298860701641108

van der Zee, K., Thijs, M., &amp; Schakel, L. (2002). The relationship of emotional intelligence with academic intelligence and the big five. ''European Journal of Personality, 16,'' 103-125. doi: 10.1002/per.434

von Wyl, A., Perren, S., Braune-Krickan, K., Simoni, H., Stadlmayr, W., Burgin, D., &amp; von Klitzing, K. (2008). How early triadic family processes predict children’s strengths and difficulties at age three. ''European Journal of Developmental Psychology, 5,'' 466-491. doi: 10.1080/17405620600989701

West, R., &amp; Turner, L. H. (2007). ''Introduction to communication theory: Analysis and application (3rd ed.).'' New York: The McGraw-Hill Companies.

Wilson, P. M., Mack, D. E., &amp; Grattan, K. P. (2008). Understanding motivation for exercise: A self-determination theory perspective. ''Canadian Psychology, 49,'' 250-256. doi: 0.1037/a0012762
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{{MECR|1=http://www.youtube.com/watch?v=Kexgk_aah5o&amp;}}
{{TOCright}}

==Overview==

[[File:Sunset Hopping.jpg|200px|left]]

[[File:In the evening sun.jpg|200px|right]]

[[File:Grass of Happiness.jpg|thumb|200px|left|Happy in nature]]

Underpinning most of what is discussed in this chapter is the evolutionary explanations of [[w:Emotion|emotions]], the [[w:Biophilia_hypothesis|Biophilia Hypothesis]] and the way nature plays an important role in our psychological well-being. 

The focus of this chapter is to learn lessons from the research into  [[#Environmental_aesthetics|environmental aesthetics]], [[#Restorative_Environment_.26_Attention_Restoration_Theory|the restorative environment]], [[#Restorative_Environment_.26_Attention_Restoration_Theory|attention restoration theory]] and [[#Nature_Deficit_Disorder |nature deficit disorder]] to help us improve our lives. [[#Seasonal_Affective_Disorder |Seasonal affective disorder]] and [[#Green_exercise|green exercise]] are discussed because of their relevance to mental illness and [[#Miscellaneous_tips_on_how_to_use_nature_to_improve_your_life|miscellaneous tips]] for improving our psychological well-being are given at the end.

Keep an eye out for the light green boxes which show research based on the theories covered. Look for the heading [[#Miscellaneous_tips_on_how_to_use_nature_to_improve_your_life|&quot;use nature to improve your life&quot;]] for tips based on both the theories and the research.

==Evolution and emotions==
People are motivated to behave in ways that have made them happy in the past and avoid behaviours which have led to them being sad (Nesse, 1989). Nesse (1989) suggests that from an evolutionary point of view, it is most useful for a person to conserve energy when the outcome is unlikely to be a good one but to use all energy stores when the effort will result in good consequences.

{| cellpadding=&quot;10&quot; cellspacing=&quot;5&quot; style=&quot;width: 100%; background-color: ForrestGreen; margin-left: auto; margin-right: auto&quot;
| style=&quot;width: 30%; background-color:Wheat; border: 3px solid YellowGreen; vertical-align: top; -moz-border-radius-topleft: 10px; -moz-border-radius-bottomleft: 10px; -moz-border-radius-topright: 10px; -moz-border-radius-bottomright: 10px;&quot; &lt;!--rowspan=&quot;2&quot;--&gt; |&lt;div style=&quot;text-align: center&quot;&gt;&lt;p style =&quot;line-height:20px;&quot;&gt;&lt;big&gt;“Emotions are specialized modes of operation shaped by natural selection to adjust the physiological, psychological, and behavioural parameters of the organism in ways that increase its capacity and tendency to respond adaptively to the threats and opportunities characteristic of specific kinds of situations.” Nesse (1989, p. 268)&lt;/big&gt; 
&lt;br&gt;&lt;/p&gt;&lt;/div&gt;
|}

==Biophilia hypothesis==

{| cellpadding=&quot;10&quot; cellspacing=&quot;5&quot; style=&quot;width:20%; background-color: ForrestGreen; margin-left: left; margin-right: auto&quot;
| style=&quot;width: 40%; background-color:Wheat; border: 3px solid YellowGreen; vertical-align: top; -moz-border-radius-topleft: 10px; -moz-border-radius-bottomleft: 10px; -moz-border-radius-topright: 10px; -moz-border-radius-bottomright: 10px;&quot; &lt;!--rowspan=&quot;2&quot;--&gt; |&lt;div style=&quot;text-align: center&quot;&gt;&lt;p style =&quot;line-height:20px;&quot;&gt;&lt;big&gt;'''Please do this before reading on about the Biophilia Hypothesis'''&lt;/big&gt;
Which picture from each pair do you prefer?&lt;/p&gt;
&lt;div style=&quot;column-count:3;-moz-column-count:3;-webkit-column-count:3&quot;&gt;
&lt;quiz display=simple&gt;
{ 1st Pair 
|type=&quot;[]&quot;}
+ [[Image:Field Hamois Belgium Luc Viatour.jpg|200px]]
- [[Image:Wasteland - geograph.org.uk - 417442.jpg|200px]]

{ 2nd Pair
|type=&quot;[]&quot;}
+ [[Image:Pachaug Trail - Porter Pond, Sterling, CT.jpg|200px]]
- [[Image:EAP-lobby-sw.jpg|200px]]

{ 3rd Pair
|type=&quot;[]&quot;}

- [[Image:Stahlbau Pichler Maciachini Mailand.jpg|200px]]
+ [[Image:41 Cherry Orchard Road.JPG|200px]]
&lt;/quiz&gt;
&lt;/div&gt;
This quiz examines preferences and introduces the topic. There are no right or wrong answers.
&lt;/div&gt;
|}

You may be asking, what has [[w:Evolution|evolution]] and emotion have to do with nature?  We are beings with a fundamental need and desire to be affiliated with nature, according to the [[w:Biophilia_hypothesis |Biophilia Hypothesis]] (Khan, 1997; Wilson, 1984). This desire stems from our ancient history where we depended on the land and lived at the mercy of Mother Nature for almost 2 million years on the East African savannahs (Kahn, 1997). Observing the landscape to find bodies of water and food sources while keeping an eye out for predators was a part of daily life and is still ingrained in our subconscious minds.

Psychological research has found that people have a preference for natural environments and built environments with natural features over built environments without natural features (Kaplan &amp; Kaplan, 1989). This will be discussed further in environmental aesthetics and attention restoration theory, as they are both related to our general preference for natural environments and the benefits of nature to our well-being. 

The [[w:Biophilia_hypothesis|biophilia hypothesis]] suggests that the long history between humans and nature shaped our cognitions and emotions in a way that the modern human is now wired for an appreciation and evaluation of natural environments (Gullone, 2000). Those humans who had a genotype best for making behavioural responses most likely to enhance survival and reproduction were kept in the population through natural selection (Gullone, 2000). Behaviours in response to natural stimulation are either approach or avoidance behaviours. These adaptive behaviours are called biophilia and biophobia. Biophilia is the approach behaviour towards natural environments judge as fit for survival. Biophobia is the aversion and avoidance of potential dangers such as spiders, snakes and bears (Gullone, 2000). These behaviours are inherited from one generation to another and even when humans stopped living in a natural environment these rules, behaviours and stimuli responses did not disappear.

'''Use nature to improve your life'''

There is quite obviously a gap between today’s 21st century living habitats and that of our ancient ancestors. It has been suggested that reducing this gap and using the natural environmental is a good way to improve our psychological well-being and the quality of life (Buss, 2000). Being in touch with nature is one way to make you happy. It has been recommended that the differences between modern and ancestral living conditions be minimised and a [[w:Paleolithic_lifestyle|Paleolithic lifestyle]] to be taken up to increase psychological well-being (Buss, 2000). It is also recommended to develop deeper friendships rather than fair weather friends which reflects the society of our ancestors (Buss, 2000). These are some examples of how to keep ourselves close to our evolutionary roots and minimise the gap between now and then.

== Environmental aesthetics==
[[File:GoldenMedows.jpg|thumb|left|200px|Aesthetically Pleasing]]

Environmental preference studies aim to determine the aesthetic value of a given stimuli using questionnaires, images and physical environments (Galindo &amp; Rodriguez, 2000). There are various questionnaires that have been employed to measure the various components of environmental aesthetics. Descriptive scales have been used for describing the physical attributes in an environment or image. Affective scales measure moods and reactions to the stimuli. Appraisal scales indicate the aesthetic value of the settings (Galindo &amp; Rodriguez, 2000). Many studies have concluded that people prefer park-like settings with short grass, mature trees and water (Balling &amp; Falk, 1982). This indicates that we are inherently predisposed to judge [[w:Savanna|savannah]] like environments as more aesthetically pleasing than others. Such results support the biophilia hypothesis. 

The work of Daniel E. Berlyne explores the scientific study of aesthetic behaviour with a specific focus on the “appreciator” of the creation being viewed. His theory is developed in the evolutionist framework. He suggests that aesthetics play an important function in the survival of the human species. It is beneficial for humans to have an ability to detect aesthetically pleasing environments than to lack the ability. Natural selection would theoretically reduce the number of people in the population who did not possess these necessary abilities.  

The work of [[w:Rachel_and_Stephen_Kaplan|Rachel and Stephen Kaplan]] builds on that of Berlyne. Kaplan and Kaplan were concerned with the basic cognitive needs that people have with regard to their physical environments (Kaplan &amp; Kaplan, 1989). These needs include the need to make sense of the environment and its features while extracting useful information. The second is the need to be involved in the environment (Galindo &amp; Rodriguez, 2000). Environments that meet these needs generate responses of attraction and aesthetic preference in all individuals studied (Kaplan &amp; Kaplan, 1989). An [[w:Evolutionist|evolutionist]] would say that avoiding environments that don't meet needs while approaching those which do, is a desirable and adaptive ability (Galindo &amp; Rodriguez, 2000).

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&lt;big&gt;Aesthetics, Pleasure and Arousal: Galindo &amp; Rodriguez, 2000&lt;/big&gt;

{|border=&quot;1&quot; cellpadding=&quot;5&quot; cellspacing=&quot;0&quot; align=&quot;right&quot;
|-
! scope=&quot;col&quot; style=&quot;background:#F5DEB3;&quot; | 
! scope=&quot;col&quot; style=&quot;background:#9ACD32;&quot; | High Pleasure
! scope=&quot;col&quot; style=&quot;background:#9ACD32;&quot; | Low Pleasure
|-
! scope=&quot;row&quot; style=&quot;background:#9ACD32;&quot; | High Arousal 
| Exciting || Distress
|-
! scope=&quot;row&quot; style=&quot;background:#9ACD32;&quot; | Low Arousal 
|| Tranquillity|| Boring
|}
 
Two independent and bipolar dimensions, pleasure and arousal, are used to describe the moods and affective responses that a person feels when observing an environment (Galindo &amp; Rodriguez, 2000). The table  demonstrates the four possible outcomes of when the two dimensions are low or high. 

The Galindo and Rodriguez (2000) study investigated the relationship between aesthetic judgements and affective responses/psychological well-being.  Aesthetic appraisal responses were closely associated with arousal and pleasure, indicating that there is a relationship between environmental aesthetics and mood. 
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'''Use nature to improve your life''' 

This knowledge of human’s preferences for nature over human-made environments has many applications. For example, in a hospital ward there was a high sick-leave rate in staff before the installation of full spectrum daylight bulbs and green plants in 80m&lt;sup&gt;2&lt;/sup&gt; on the premises. After the environmental aesthetic changes were made, there was a 32% reduction in tiredness, 25% drop in the sick-leave rate, 45% less headaches and 31% reduction in sore throats (Fjeld, 1998a, b). Are you a manager or boss in a company where you would like to reduce the sick-leave rate? How about investing in some green plants and [[w:Full-spectrum_light|full-spectrum]] daylight bulbs for the workplace and watch the way your employees react?

==Restorative environment &amp; Attention Restoration Theory==

Mental fatigue is the feeling of being worn out after working too hard for too long and feeling the need for a well-earned break. Everybody has experienced mental fatigue at some point, but it is interesting to note that you can feel mentally fatigued from working hard on projects you enjoy (Kaplan &amp; Kaplan, 1989). Mental fatigue is often accompanied with the inability to focus and keep our attention on the project we are working on, instead distraction takes place and our minds wander. 

William James (1892) suggested that there are two types of attention; directed and involuntary.  Involuntary attention is the type that doesn’t require any effort and is often used when there is something interesting to look at (Kaplan &amp; Kaplan, 1989). Directed attention on the other hand is that which requires effort and drains our minds, such as a work project or an essay (Kaplan &amp; Kaplan, 1989). Sleep can provide relief from this mental fatigue but the restorative experience is often more convenient. 

The restoration experience involves restoring our stores of directed attention while increasing our mental well-being by using involuntary attention. We can then return refreshed to do those things that need to be done.  Attention Restoration Theory (Kaplan &amp; Kaplan, 1989) proposes that a restorative experience involves four components. These are being away, the extent to which it feels like being in another ‘world’, fascination, and compatibility.

[[File:Vrijbroekpark - by frank wouters.jpg|thumb|left|Restorative Experience]]

'''Use nature to improve your life''' 

'''Being away'''

Being away involves stopping mental effort and directed attention, escaping from distractions and from the work that is normally done. One does not need to be a long way from the workplace to escape from these things. An environment need only be distinctive and separate from the workplace. Perhaps you could try escaping to the nearby park during the lunch break to your eat lunch and while you are there, watch the birds and animals in the park?

'''Extent of other worldliness'''

The extent to which the environment is different to your usual place is important because without the distinctiveness it will not be a restorative experience. A place that can be described as a ‘whole other world’ is an important property of the environment to which you escape to. A garden where there is more to explore and see in the garden than what one sees at first sight is another important part of extent. Kaplan and Kaplan (1989) suggest that Japanese gardens are the best for this as the miniature effect creates a sense of distinctiveness and uniqueness. If you are in the process of landscaping, consider a garden that encourages exploration using hidden features and overgrown pathways.

'''Fascination'''

Fascination is the next key to the restorative experience to help you regain mental well-being. Fascination involves involuntary attention and therefore keeps the person from getting bored but also there is no effort needed for attention to take place. Wild animals, sunsets, waterfalls, caves, fires, stars, clouds can all grab our involuntary and effortless attention if we let them and are called soft fascinations. They should not hold our attention for too long (otherwise they become too distracting to pull ourselves out) and they should be aesthetically pleasing. Go and watch the clouds go by on a day when you’ve had enough of work. At night watch the stars in your backyard. These forms of soft fascinations will allow your directed attention to have a break and its stores to refill while your involuntary attention takes over for a while.

'''Compatibility'''

Compatibility and relatedness come back to the evolutionary perspective and the biophilia hypothesis. That is, that on an innate and inherited level we as humans prefer natural environments to human-made environments and we feel a sense of awe and wonder when in its presence. Relatedness and compatibility can occur in these natural environments. Psychological well-being occurs when we are in a compatible environment. Being in an incompatible environment requires a lot of effort and attention. Next time you are finding it hard to read a difficult text due to distractions at work, head to a park bench and immerse yourself in a compatible restorative environment. 

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&lt;big&gt;Restorative environments through a hospital window (Ulrich, 1984)&lt;/big&gt;

The recovery records of 46 cholecystectomy (gall bladder surgery) patients from a Pennsylvania hospital were examined. The aim was to explore the difference in recovery rates of those with a window facing a brick wall and those who had a view of foliage. The patients were grouped into 23 pairs based on sex, age, smoking habits, and weight and floor level. Results for the months between 1st of May and October 20th for the years 1972 to 1981 were used, because as during these months the trees had foliage. The length of hospitalisation for those with a tree-view was 7.96 days while those with a view of a brick wall stayed in for 8.70 days. Nurses’ notes on the wall-view patients indicated that they were in a more negative mood than those in the tree-view room, although it was not statistically significant. Those in the tree-view room took fewer moderate to strong pain killers than the wall-view patients in the 2-5 days after surgery.  Ulrich noted that the brick wall view was quite hideous and boring. He suggests that a view of another built environment, such as a busy street, may produce different results because the patient is more stimulated. The results of the 1984 study suggest that natural environments do indeed provide a restorative environment which improves both physical and mental well-being. 
{{RoundBoxBottom}}

==Nature Deficit Disorder==

[[File:Eye to Eye.jpg|thumb|left|Children with Nature]]

Author of “[[w:Last_Child_in_the_Woods|Last Child In the Woods]]”, [[w:Richard Louv| Richard Louv]] has suggested that the children of today’s society are missing out on vital interaction with nature. [[w:Nature deficit disorder|Nature Deficit Disorder]], as Louv puts it, is the cause of the rise in obesity, attention disorders and depression. Nature Deficit Disorder is not recognised as an official disorder, however many of the points Louv raises relate to the current chapter. The book is recommended for parents who are worried about the health of their children and their lack of direct exposure to nature. 

The main features of Louv’s Nature Deficit Disorder theory are based around the relationship between children and nature. He suggests that there is a new relationship between child and nature in which children do not interact with the natural world as much as they should. Secondly, he suggests that we all need direct exposure to nature in our lives and without it mental fatigue, irritability and agitation creep in.  Thirdly, Louv suggests that busy schedules and fear keep our children from enjoying nature. Thankfully his book “[[w:Last_Child_in_the_Woods|Last Child In the Woods]]” also provides ways to reunite children and nature (see tips for more).

Please keep in mind that this disorder is currently not in the [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders |DSM]] and therefore not accepted by general psychiatry. Further research and theory development is needed before acceptance as a mental disorder. 

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&lt;big&gt;Vegetation and children: Wells and Evans (2003)&lt;/big&gt;

Wells and Evans collected data from 337 rural children from five small towns in upstate New York. They were in grades 3 to 5 with an average age just over nine years. Parents reported the psychological distress shown by their children on the Rutter Child Behaviour Questionnaire. Children provided their own ratings of global self-worth on the Harter Competency Scale. The authors explored the impact of nearby nature on children subject to a stressful life event. To assess the frequency of stressful life events, the Lewis Stressful Life Events Scale was used.

As expected, exposure to stressful life events correlated with psychological distress. Secondly, nearby nature was found to be a moderator between stressful life events and psychological distress.  There was a large difference between the psychological distress of children who had low exposure to nature and those who had high exposure to nature. Those with high exposure to nature had less distress than those with low exposure to nature even though they had the same levels of stressful life events. The effect was particularly large when there were high levels of stressful life events.  
{{RoundBoxBottom}}

'''Use nature to improve your life''' 

Children who experience high levels of stressful life events would benefit from living nearby nature. If this is not an option, consider investing in a veggie patch or building a cubby house in your backyard. Whether you believe your child is distressed or not, encourage your children to be active in the outdoors. If you have a busy schedule, consider allocating time to throwing a ball around outside with your child. It will be good for you as well as your child!

== Seasonal Affective Disorder ==

[[File:Schneelandschaft Furx.JPG|thumb|A winter without Sun]]

Biological changes occur on a continuum throughout the year and while many people may feel lethargic and irritable in the colder and darker months of the year (Eagles, 2003), on the extreme end of this continuum is [[w:Seasonal_affective_disorder|Seasonal Affective Disorder]] which impairs the functioning of those who suffer with it.

Seasonal Affective Disorder (SAD) is a disorder in the [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders|DSM-IV]] that presents as major depression with a seasonal pattern (Targum &amp; Rosenthal, 2008). People with SAD slow down and lose energy in the darker months of the year and often have trouble waking in the morning. There can also be problems with concentration and weight gain which can be caused from excessive eating of sweets and other starchy foods (Targum &amp; Rosenthal, 2008).

SAD is most common in the winter months where there is relatively little sun (Rosenthal, 2009). Severe SAD can cause major depressive symptoms on cloudy days (Targum &amp; Rosenthal, 2008). Research has suggested that patients with SAD can have more severe symptoms than clients with non-seasonal major depression who have attempted suicide (Pendse, Engstroem &amp; Traeskman-Bendz, 2004). Treatment is necessary for those who have SAD.

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&lt;big&gt;Melatonin, Bright Light Therapy and Dawn Simulation: Avery et al, 2001; Terman et al, 2001.&lt;/big&gt; 

[[w:Melatonin|Melatonin]] is a natural hormone which is used by all mammals to indicate that darkness is approaching and that it is time for rest. Research has suggested that the longer hours of darkness cause an imbalance to the melatonin rhythm and this is why people become more lethargic in the darker months (Brown, Pandi-Perumal, Trakht &amp; Cardinali, 2010).

Bright light therapy and dawn simulation both attempt to fix the melatonin imbalance and rhythm and restore proper sleep and mood ((Avery, Eder, Bolte, Hellekson, Dunner, Vitiell &amp; Prinz, 2001; Terman, Terman &amp; Cooper, 2001). To help reset the melatonin rhythm, bright light treatment is best administered in the morning eight and a half hours after melatonin onset(Terman et al., 2001). Bright light therapy has been criticised for the inconvenience caused to the patient.

Dawn simulation has been suggested as a better for the treatment of SAD (Avery et al, 2001). Dawn simulation is the use of dim lights slowly increasing in intensity until the patient wakes up and simulates the rising of the sun in the morning (Avery et al., 2001). Compliance of the patient is greater with dawn simulation than bright light therapy because it saves time and is very convenient (Avery et al., 2001).
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'''Use nature to improve your life''' 

Having knowledge of SAD can help you identify if this applies to you and what natural light treatments are available. Read &quot;Winter Blues: Everything You Need to Know to Beat Seasonal Affective Disorder&quot; by [[w:Norman_E._Rosenthal|Norman Rosenthal]] for more information. You should consult your doctor or psychologist for further information and diagnosis of SAD.

==Green exercise==
[[File:What is green exercise.svg|thumb|left|Table 1. ''Components of Green Exercise'']] 

Physical activity and exposure to nature both have positive effects on psychological well-being and physical health (Pretty, Peacock, Sellens &amp; Griffin, 2005). Table 1 shows the components that define [[w:Green_exercise|green exercise]]. Combining physical exercise and exposure to nature it termed green exercise and may promote even better physical and mental health than either one alone. [[Green_exercise]] incorporates environmental aesthetics and restorative environments. Engagement with nature can be seen on three levels (Mackay &amp; Neill, 2010): viewing nature, being in the presence of nature, and being involved in nature. Green exercise is about being involved in nature. Gardening, jogging outdoors, watersports and mountain biking can be considered as green exercise. 

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&lt;big&gt;Exercise and projected scenes: Pretty et al. 2005&lt;/big&gt;

{| border=&quot;1&quot; cellpadding=&quot;5&quot; cellspacing=&quot;0&quot; align=&quot;right&quot;
|-
! scope=&quot;col&quot; style=&quot;background:#9ACD32;&quot;|Condition
! scope=&quot;col&quot; style=&quot;background:#9ACD32;&quot;|Example
|-
!scope=&quot;col&quot; style=&quot;background:#F5DEB3;&quot; | Control
| No Image 
|-
!scope=&quot;col&quot; style=&quot;background:#F5DEB3;&quot; |Rural Pleasant
| Trees, fields etc 
|-
!scope=&quot;col&quot; style=&quot;background:#F5DEB3;&quot; |Urban Pleasant
| Cityscape with natural features ie city parks
|-
!scope=&quot;col&quot; style=&quot;background:#F5DEB3;&quot; |Rural Unpleasant
| Broken down cars in the bush etc
|-
!scope=&quot;col&quot; style=&quot;background:#F5DEB3;&quot; |Urban Unpleasant
| Broken windows, wire fencing etc 
|-
|}

There were five conditions with 20 participants. In four of the conditions 30 scenes were projected onto a wall whilst the participants exercised on a treadmill.  Blood pressure, self-esteem and mood were measured before and after the exercise. 

Results indicated that in the control group exercise reduced blood pressure and improved self-esteem and mood. When exercise was paired with pleasant scenes the mood and self-esteem of participants was higher than in other conditions. Unpleasant scenes when paired with exercise reduced the effect of exercise on self-esteem and self-esteem was lower in these conditions.
{{RoundBoxBottom}}

'''Use nature to improve your life'''
[[File:Mountain bike ParcoSibillini.jpg|thumb|Green exercise]]

Next time you hop on your treadmill, consider facing it towards a window where you can see outside. Or go one step further and go for a jog outside, go hiking or go canoeing on the local lake. Being motivated to exercise can be hard, so check out the [[Motivation and emotion/Book/Exercise motivation|Exercise Motivation chapter]] for some tips.

== Miscellaneous tips on how to use nature to improve your life==

* Give flowers to a woman. When you do she will produce a [[w:Smile#Duchenne smiling|Duchenne smile]] and will be in a more positive mood for 3 days. (Haviland-Jones, Rosario, Wilson &amp; McGuire, 2005). 
* Make your backyard into a habitat for native flora and fauna. Build a bird bath and plant natives (requires less watering than non-natives and they look great).
* Go hiking and camping. Light a fire and star gaze at night (please check local fire bans beforehand).  
* Be in nature and be aware of your senses. Listen, smell, touch and watch (Louv, 2008). Try the [http://wilderdom.com/games/descriptions/SensualAwarenessInventory.html Sensual Awareness Inventory] for this activity.
* Plant a veggie garden. Good for saving money on produce. Great for the kids to see where their food comes from and be involved in the process (Louv, 2008).

==Summary==
Lessons about nature and psychological well-being can be learnt from the research and theories of environmental and evolutionary schools of psychology.  This chapter was written to help you realise that nature plays a vital role in our psychological well-being.  You now know that: 
* You can use restorative environments to reduce mental fatigue. (Attention Restoration Theory) 
* Being in touch with nature and our ancestral roots can improve your life. (Environmental Aesthetics &amp; Biophilia Hypothesis)
* Children need nature for their mental health. (Nature Deficit Disorder) 
* Emotions can be dramatically affected by the change of seasons. (Seasonal Affective Disorder)
* Combining exercise with nature is best. (Green exercise) 

The take away message from the chapter is this: '''go outdoors and use nature to improve your life'''.

==See also==
* [http://www.acfonline.org.au/news-media/acf-opinion/ten-reasons-why-we-need-more-contact-nature Ten reasons why we need more contact with nature] 
* [[Motivation and emotion/Book/Animals and emotion|Animals and emotion: How connection with animals can improve emotion and well-being]] 
* [[Motivation and emotion/Book/Weather and emotion | Weather and Emotion: How the weather effects our emotions]]
* [[/Transcript/|Transcript for the 5 minute audiovisual chapter overview]]

== References ==
{{Hanging indent|
Avery, D.H., Eder, D.N., Bolte, M.A., Hellekson, C.J, Dunner,D.L., Vitiello, M.V., &amp; Prinz, P.N. (2001). Dawn simulation and bright light in the treatment of sad: A controlled study. ''Society of Biological Psychiatry'', ''50'', 205-216.

Balling, J.D. &amp; Falk, J.H. (1982). Development of visual preference for natural environments. ''Environment and Behaviour'', ''14''(1), 5-28. 

Brown, G.M., Pandi-Perumal, S.R., Trakht &amp; Cardinali,. (2010). The role of melatonin in seasonal affective disorder. In T. Partonen, &amp; S.R Pandi-Perumal (Eds.), ''Seasonal Affective Disorder: Practice and research'' (pp. 149-162). New York, NY: Oxford University Press.

Buss, D.M. (2000). The evolution of happiness. ''American Psychologist'', ''55''(1), 15-23. 

Eagles, J.M. (2003). Seasonal affective disorder. ''The British Journal of Psychiatry'', ''182'', 174-176.

Fjeld, T. (1998a). Plants in interior surroundings—a source to health. ''gartneryrket 13''(15). 

Fjeld, T. (1998b). Plants; Light; Interior and health. ''research report. the norwegian radium hospital Olso''

Galindo, P. &amp; Rodriguez, J.A. (2000). Environmental aesthetics and psychological well-being: Relationships between preferences judgements for urban landscapes and other relevant affective responses. ''Psychology in Spain, 4'' (1), 13-27.

Gullone, E. (2000). The biophilia hypothesis and life in the 21st century: Increasing mental health or increasing pathology?. ''Journal of Happiness Studies, 1'', 293-321.

Haviland-Jones, J., Rosario, H.H., Wilson, P., &amp; McGuire, T.R. (2005). An environmental approach ot positive emotion: Flowers. ''Evolutionary Psychology, 3,''  104-132. 

James, W. (1892). ''Psychology: The briefer course''. New York: Holt.  

Kaplan, R. &amp; Kaplan, S. (1989). ''The experience of nature: A psychological perspective.'' Cambridge: Cambridge University Press. 

Khan, P.H. (1997). Developmental psychology and the biophilia hypothesis: Children’s affiliation with nature. ''Developmental Review, 17,'' 1-61.

Louv, R. (2008). ''Last Child in the Woods: Saving our children from nature-deficit disorder.'' Chapel Hill, NC: Algonoquin Books.

Mackay, G.J. &amp; Neill, J.T. (2010). The effect of green exercise on state anxiety and the role of exercise duration, intensity and greenness: a quasi-experimental study. ''Psychology of Sport and Exercise, 11,'' 238-245.

Nesse, R.M. (1989). Evolutionary explanations of emotions. ''Human Nature, 1'' (3). 261-289.

Pendse, B.P., Engstroem, G., &amp; Traeskman-Bendz, L. (2004) Psychopathology of seasonal affective disorder patients in comparison with major depression patients who have attempted suicide. ''Journal of Clinical Psychiatry 65'', 322–327.

Pretty, J.,  Peacock, J., Sellens, M., &amp; Griffin, M. (2005). The mental and physical health outcomes of green exercise. ''International Journal of Environmental Health Research, 15'' (5). 319-337.

Rosenthal, N. (2009). Issues for DSM-V: Seasonal affective disorder and seasonality. ''American Journal of Psychiatry, 166'' (8), 52-53.

Targum, S.D. &amp; Rosenthal, N. (2008). Research to practice; Seasonal affective disorder. ''Psychiatry'', May,31-33.

Terman, J., Terman, M., Lo, E., &amp; Cooper, T. (2001). Circadian time of morning light administration and therapeutic response in winter depression. ''Archives of General Psychiatry, 58,'' 69-75.

Ulrich, R.S. (1984). View through a window may influence recovery from surgery. ''Science, 224.'' 420-421.  

Wells, N.M. &amp; Evans, G.W. (2003). Nearby nature: A buffer of life stress among rural children. ''Environment and Behaviour, 35''(3), 311-330. 

Wilson, E. O. (1984). ''Biophilia''. Cambridge: Harvard University Press.}}

==External links==

* [http://www.thehappinessinstitute.com/blog/article.aspx?c=3&amp;a=1987 The Happiness Institute: Happiness and Nature]
* [http://ecco.vub.ac.be/?q=node/127 Evolutionary Well-Being: the paleolithic model]

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==Overview==

[[File:Sunset Hopping.jpg|200px|left]]

[[File:In the evening sun.jpg|200px|right]]

[[File:Grass of Happiness.jpg|thumb|200px|left|Happy in nature]]

Underpinning most of what is discussed in this chapter is the evolutionary explanations of [[w:Emotion|emotions]], the [[w:Biophilia_hypothesis|Biophilia Hypothesis]] and the way nature plays an important role in our psychological well-being. 

The focus of this chapter is to learn lessons from the research into  [[#Environmental_aesthetics|environmental aesthetics]], [[#Restorative_Environment_.26_Attention_Restoration_Theory|the restorative environment]], [[#Restorative_Environment_.26_Attention_Restoration_Theory|attention restoration theory]] and [[#Nature_Deficit_Disorder |nature deficit disorder]] to help us improve our lives. [[#Seasonal_Affective_Disorder |Seasonal affective disorder]] and [[#Green_exercise|green exercise]] are discussed because of their relevance to mental illness and [[#Miscellaneous_tips_on_how_to_use_nature_to_improve_your_life|miscellaneous tips]] for improving our psychological well-being are given at the end.

Keep an eye out for the light green boxes which show research based on the theories covered. Look for the heading [[#Miscellaneous_tips_on_how_to_use_nature_to_improve_your_life|&quot;use nature to improve your life&quot;]] for tips based on both the theories and the research.

==Evolution and emotions==
People are motivated to behave in ways that have made them happy in the past and avoid behaviours which have led to them being sad (Nesse, 1989). Nesse (1989) suggests that from an evolutionary point of view, it is most useful for a person to conserve energy when the outcome is unlikely to be a good one but to use all energy stores when the effort will result in good consequences.

{| cellpadding=&quot;10&quot; cellspacing=&quot;5&quot; style=&quot;width: 100%; background-color: ForrestGreen; margin-left: auto; margin-right: auto&quot;
| style=&quot;width: 30%; background-color:Wheat; border: 3px solid YellowGreen; vertical-align: top; -moz-border-radius-topleft: 10px; -moz-border-radius-bottomleft: 10px; -moz-border-radius-topright: 10px; -moz-border-radius-bottomright: 10px;&quot; &lt;!--rowspan=&quot;2&quot;--&gt; |&lt;div style=&quot;text-align: center&quot;&gt;&lt;p style =&quot;line-height:20px;&quot;&gt;&lt;big&gt;“Emotions are specialized modes of operation shaped by natural selection to adjust the physiological, psychological, and behavioural parameters of the organism in ways that increase its capacity and tendency to respond adaptively to the threats and opportunities characteristic of specific kinds of situations.” Nesse (1989, p. 268)&lt;/big&gt; 
&lt;br&gt;&lt;/p&gt;&lt;/div&gt;
|}

==Biophilia hypothesis==

{| cellpadding=&quot;10&quot; cellspacing=&quot;5&quot; style=&quot;width:20%; background-color: ForrestGreen; margin-left: left; margin-right: auto&quot;
| style=&quot;width: 40%; background-color:Wheat; border: 3px solid YellowGreen; vertical-align: top; -moz-border-radius-topleft: 10px; -moz-border-radius-bottomleft: 10px; -moz-border-radius-topright: 10px; -moz-border-radius-bottomright: 10px;&quot; &lt;!--rowspan=&quot;2&quot;--&gt; |&lt;div style=&quot;text-align: center&quot;&gt;&lt;p style =&quot;line-height:20px;&quot;&gt;&lt;big&gt;'''Please do this before reading on about the Biophilia Hypothesis'''&lt;/big&gt;
Which picture from each pair do you prefer?&lt;/p&gt;
&lt;div style=&quot;column-count:3;-moz-column-count:3;-webkit-column-count:3&quot;&gt;
&lt;quiz display=simple&gt;
{ 1st Pair 
|type=&quot;[]&quot;}
+ [[Image:Field Hamois Belgium Luc Viatour.jpg|200px]]
- [[Image:Wasteland - geograph.org.uk - 417442.jpg|200px]]

{ 2nd Pair
|type=&quot;[]&quot;}
+ [[Image:Pachaug Trail - Porter Pond, Sterling, CT.jpg|200px]]
- [[Image:EAP-lobby-sw.jpg|200px]]

{ 3rd Pair
|type=&quot;[]&quot;}

- [[Image:Stahlbau Pichler Maciachini Mailand.jpg|200px]]
+ [[Image:41 Cherry Orchard Road.JPG|200px]]
&lt;/quiz&gt;
&lt;/div&gt;
This quiz examines preferences and introduces the topic. There are no right or wrong answers.
&lt;/div&gt;
|}

You may be asking, what has [[w:Evolution|evolution]] and emotion have to do with nature?  We are beings with a fundamental need and desire to be affiliated with nature, according to the [[w:Biophilia_hypothesis |Biophilia Hypothesis]] (Khan, 1997; Wilson, 1984). This desire stems from our ancient history where we depended on the land and lived at the mercy of Mother Nature for almost 2 million years on the East African savannahs (Kahn, 1997). Observing the landscape to find bodies of water and food sources while keeping an eye out for predators was a part of daily life and is still ingrained in our subconscious minds.

Psychological research has found that people have a preference for natural environments and built environments with natural features over built environments without natural features (Kaplan &amp; Kaplan, 1989). This will be discussed further in environmental aesthetics and attention restoration theory, as they are both related to our general preference for natural environments and the benefits of nature to our well-being. 

The [[w:Biophilia_hypothesis|biophilia hypothesis]] suggests that the long history between humans and nature shaped our cognitions and emotions in a way that the modern human is now wired for an appreciation and evaluation of natural environments (Gullone, 2000). Those humans who had a genotype best for making behavioural responses most likely to enhance survival and reproduction were kept in the population through natural selection (Gullone, 2000). Behaviours in response to natural stimulation are either approach or avoidance behaviours. These adaptive behaviours are called biophilia and biophobia. Biophilia is the approach behaviour towards natural environments judge as fit for survival. Biophobia is the aversion and avoidance of potential dangers such as spiders, snakes and bears (Gullone, 2000). These behaviours are inherited from one generation to another and even when humans stopped living in a natural environment these rules, behaviours and stimuli responses did not disappear.

'''Use nature to improve your life'''

There is quite obviously a gap between today’s 21st century living habitats and that of our ancient ancestors. It has been suggested that reducing this gap and using the natural environmental is a good way to improve our psychological well-being and the quality of life (Buss, 2000). Being in touch with nature is one way to make you happy. It has been recommended that the differences between modern and ancestral living conditions be minimised and a [[w:Paleolithic_lifestyle|Paleolithic lifestyle]] to be taken up to increase psychological well-being (Buss, 2000). It is also recommended to develop deeper friendships rather than fair weather friends which reflects the society of our ancestors (Buss, 2000). These are some examples of how to keep ourselves close to our evolutionary roots and minimise the gap between now and then.

== Environmental aesthetics==
[[File:GoldenMedows.jpg|thumb|left|200px|Aesthetically Pleasing]]

Environmental preference studies aim to determine the aesthetic value of a given stimuli using questionnaires, images and physical environments (Galindo &amp; Rodriguez, 2000). There are various questionnaires that have been employed to measure the various components of environmental aesthetics. Descriptive scales have been used for describing the physical attributes in an environment or image. Affective scales measure moods and reactions to the stimuli. Appraisal scales indicate the aesthetic value of the settings (Galindo &amp; Rodriguez, 2000). Many studies have concluded that people prefer park-like settings with short grass, mature trees and water (Balling &amp; Falk, 1982). This indicates that we are inherently predisposed to judge [[w:Savanna|savannah]] like environments as more aesthetically pleasing than others. Such results support the biophilia hypothesis. 

The work of Daniel E. Berlyne explores the scientific study of aesthetic behaviour with a specific focus on the “appreciator” of the creation being viewed. His theory is developed in the evolutionist framework. He suggests that aesthetics play an important function in the survival of the human species. It is beneficial for humans to have an ability to detect aesthetically pleasing environments than to lack the ability. Natural selection would theoretically reduce the number of people in the population who did not possess these necessary abilities.  

The work of [[w:Rachel_and_Stephen_Kaplan|Rachel and Stephen Kaplan]] builds on that of Berlyne. Kaplan and Kaplan were concerned with the basic cognitive needs that people have with regard to their physical environments (Kaplan &amp; Kaplan, 1989). These needs include the need to make sense of the environment and its features while extracting useful information. The second is the need to be involved in the environment (Galindo &amp; Rodriguez, 2000). Environments that meet these needs generate responses of attraction and aesthetic preference in all individuals studied (Kaplan &amp; Kaplan, 1989). An [[w:Evolutionist|evolutionist]] would say that avoiding environments that don't meet needs while approaching those which do, is a desirable and adaptive ability (Galindo &amp; Rodriguez, 2000).

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&lt;big&gt;Aesthetics, Pleasure and Arousal: Galindo &amp; Rodriguez, 2000&lt;/big&gt;

{|border=&quot;1&quot; cellpadding=&quot;5&quot; cellspacing=&quot;0&quot; align=&quot;right&quot;
|-
! scope=&quot;col&quot; style=&quot;background:#F5DEB3;&quot; | 
! scope=&quot;col&quot; style=&quot;background:#9ACD32;&quot; | High Pleasure
! scope=&quot;col&quot; style=&quot;background:#9ACD32;&quot; | Low Pleasure
|-
! scope=&quot;row&quot; style=&quot;background:#9ACD32;&quot; | High Arousal 
| Exciting || Distress
|-
! scope=&quot;row&quot; style=&quot;background:#9ACD32;&quot; | Low Arousal 
|| Tranquillity|| Boring
|}
 
Two independent and bipolar dimensions, pleasure and arousal, are used to describe the moods and affective responses that a person feels when observing an environment (Galindo &amp; Rodriguez, 2000). The table  demonstrates the four possible outcomes of when the two dimensions are low or high. 

The Galindo and Rodriguez (2000) study investigated the relationship between aesthetic judgements and affective responses/psychological well-being.  Aesthetic appraisal responses were closely associated with arousal and pleasure, indicating that there is a relationship between environmental aesthetics and mood. 
{{RoundBoxBottom}}

'''Use nature to improve your life''' 

This knowledge of human’s preferences for nature over human-made environments has many applications. For example, in a hospital ward there was a high sick-leave rate in staff before the installation of full spectrum daylight bulbs and green plants in 80m&lt;sup&gt;2&lt;/sup&gt; on the premises. After the environmental aesthetic changes were made, there was a 32% reduction in tiredness, 25% drop in the sick-leave rate, 45% less headaches and 31% reduction in sore throats (Fjeld, 1998a, b). Are you a manager or boss in a company where you would like to reduce the sick-leave rate? How about investing in some green plants and [[w:Full-spectrum_light|full-spectrum]] daylight bulbs for the workplace and watch the way your employees react?

==Restorative environment &amp; Attention Restoration Theory==

Mental fatigue is the feeling of being worn out after working too hard for too long and feeling the need for a well-earned break. Everybody has experienced mental fatigue at some point, but it is interesting to note that you can feel mentally fatigued from working hard on projects you enjoy (Kaplan &amp; Kaplan, 1989). Mental fatigue is often accompanied with the inability to focus and keep our attention on the project we are working on, instead distraction takes place and our minds wander. 

William James (1892) suggested that there are two types of attention; directed and involuntary.  Involuntary attention is the type that doesn’t require any effort and is often used when there is something interesting to look at (Kaplan &amp; Kaplan, 1989). Directed attention on the other hand is that which requires effort and drains our minds, such as a work project or an essay (Kaplan &amp; Kaplan, 1989). Sleep can provide relief from this mental fatigue but the restorative experience is often more convenient. 

The restoration experience involves restoring our stores of directed attention while increasing our mental well-being by using involuntary attention. We can then return refreshed to do those things that need to be done.  Attention Restoration Theory (Kaplan &amp; Kaplan, 1989) proposes that a restorative experience involves four components. These are being away, the extent to which it feels like being in another ‘world’, fascination, and compatibility.

[[File:Vrijbroekpark - by frank wouters.jpg|thumb|left|Restorative Experience]]

'''Use nature to improve your life''' 

'''Being away'''

Being away involves stopping mental effort and directed attention, escaping from distractions and from the work that is normally done. One does not need to be a long way from the workplace to escape from these things. An environment need only be distinctive and separate from the workplace. Perhaps you could try escaping to the nearby park during the lunch break to your eat lunch and while you are there, watch the birds and animals in the park?

'''Extent of other worldliness'''

The extent to which the environment is different to your usual place is important because without the distinctiveness it will not be a restorative experience. A place that can be described as a ‘whole other world’ is an important property of the environment to which you escape to. A garden where there is more to explore and see in the garden than what one sees at first sight is another important part of extent. Kaplan and Kaplan (1989) suggest that Japanese gardens are the best for this as the miniature effect creates a sense of distinctiveness and uniqueness. If you are in the process of landscaping, consider a garden that encourages exploration using hidden features and overgrown pathways.

'''Fascination'''

Fascination is the next key to the restorative experience to help you regain mental well-being. Fascination involves involuntary attention and therefore keeps the person from getting bored but also there is no effort needed for attention to take place. Wild animals, sunsets, waterfalls, caves, fires, stars, clouds can all grab our involuntary and effortless attention if we let them and are called soft fascinations. They should not hold our attention for too long (otherwise they become too distracting to pull ourselves out) and they should be aesthetically pleasing. Go and watch the clouds go by on a day when you’ve had enough of work. At night watch the stars in your backyard. These forms of soft fascinations will allow your directed attention to have a break and its stores to refill while your involuntary attention takes over for a while.

'''Compatibility'''

Compatibility and relatedness come back to the evolutionary perspective and the biophilia hypothesis. That is, that on an innate and inherited level we as humans prefer natural environments to human-made environments and we feel a sense of awe and wonder when in its presence. Relatedness and compatibility can occur in these natural environments. Psychological well-being occurs when we are in a compatible environment. Being in an incompatible environment requires a lot of effort and attention. Next time you are finding it hard to read a difficult text due to distractions at work, head to a park bench and immerse yourself in a compatible restorative environment. 

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&lt;big&gt;Restorative environments through a hospital window (Ulrich, 1984)&lt;/big&gt;

The recovery records of 46 cholecystectomy (gall bladder surgery) patients from a Pennsylvania hospital were examined. The aim was to explore the difference in recovery rates of those with a window facing a brick wall and those who had a view of foliage. The patients were grouped into 23 pairs based on sex, age, smoking habits, and weight and floor level. Results for the months between 1st of May and October 20th for the years 1972 to 1981 were used, because as during these months the trees had foliage. The length of hospitalisation for those with a tree-view was 7.96 days while those with a view of a brick wall stayed in for 8.70 days. Nurses’ notes on the wall-view patients indicated that they were in a more negative mood than those in the tree-view room, although it was not statistically significant. Those in the tree-view room took fewer moderate to strong pain killers than the wall-view patients in the 2-5 days after surgery.  Ulrich noted that the brick wall view was quite hideous and boring. He suggests that a view of another built environment, such as a busy street, may produce different results because the patient is more stimulated. The results of the 1984 study suggest that natural environments do indeed provide a restorative environment which improves both physical and mental well-being. 
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==Nature Deficit Disorder==

[[File:Eye to Eye.jpg|thumb|left|Children with Nature]]

Author of “[[w:Last_Child_in_the_Woods|Last Child In the Woods]]”, [[w:Richard Louv| Richard Louv]] has suggested that the children of today’s society are missing out on vital interaction with nature. [[w:Nature deficit disorder|Nature Deficit Disorder]], as Louv puts it, is the cause of the rise in obesity, attention disorders and depression. Nature Deficit Disorder is not recognised as an official disorder, however many of the points Louv raises relate to the current chapter. The book is recommended for parents who are worried about the health of their children and their lack of direct exposure to nature. 

The main features of Louv’s Nature Deficit Disorder theory are based around the relationship between children and nature. He suggests that there is a new relationship between child and nature in which children do not interact with the natural world as much as they should. Secondly, he suggests that we all need direct exposure to nature in our lives and without it mental fatigue, irritability and agitation creep in.  Thirdly, Louv suggests that busy schedules and fear keep our children from enjoying nature. Thankfully his book “[[w:Last_Child_in_the_Woods|Last Child In the Woods]]” also provides ways to reunite children and nature (see tips for more).

Please keep in mind that this disorder is currently not in the [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders |DSM]] and therefore not accepted by general psychiatry. Further research and theory development is needed before acceptance as a mental disorder. 

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&lt;big&gt;Vegetation and children: Wells and Evans (2003)&lt;/big&gt;

Wells and Evans collected data from 337 rural children from five small towns in upstate New York. They were in grades 3 to 5 with an average age just over nine years. Parents reported the psychological distress shown by their children on the Rutter Child Behaviour Questionnaire. Children provided their own ratings of global self-worth on the Harter Competency Scale. The authors explored the impact of nearby nature on children subject to a stressful life event. To assess the frequency of stressful life events, the Lewis Stressful Life Events Scale was used.

As expected, exposure to stressful life events correlated with psychological distress. Secondly, nearby nature was found to be a moderator between stressful life events and psychological distress.  There was a large difference between the psychological distress of children who had low exposure to nature and those who had high exposure to nature. Those with high exposure to nature had less distress than those with low exposure to nature even though they had the same levels of stressful life events. The effect was particularly large when there were high levels of stressful life events.  
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'''Use nature to improve your life''' 

Children who experience high levels of stressful life events would benefit from living nearby nature. If this is not an option, consider investing in a veggie patch or building a cubby house in your backyard. Whether you believe your child is distressed or not, encourage your children to be active in the outdoors. If you have a busy schedule, consider allocating time to throwing a ball around outside with your child. It will be good for you as well as your child!

== Seasonal Affective Disorder ==

[[File:Schneelandschaft Furx.JPG|thumb|A winter without Sun]]

Biological changes occur on a continuum throughout the year and while many people may feel lethargic and irritable in the colder and darker months of the year (Eagles, 2003), on the extreme end of this continuum is [[w:Seasonal_affective_disorder|Seasonal Affective Disorder]] which impairs the functioning of those who suffer with it.

Seasonal Affective Disorder (SAD) is a disorder in the [[w:Diagnostic_and_Statistical_Manual_of_Mental_Disorders|DSM-IV]] that presents as major depression with a seasonal pattern (Targum &amp; Rosenthal, 2008). People with SAD slow down and lose energy in the darker months of the year and often have trouble waking in the morning. There can also be problems with concentration and weight gain which can be caused from excessive eating of sweets and other starchy foods (Targum &amp; Rosenthal, 2008).

SAD is most common in the winter months where there is relatively little sun (Rosenthal, 2009). Severe SAD can cause major depressive symptoms on cloudy days (Targum &amp; Rosenthal, 2008). Research has suggested that patients with SAD can have more severe symptoms than clients with non-seasonal major depression who have attempted suicide (Pendse, Engstroem &amp; Traeskman-Bendz, 2004). Treatment is necessary for those who have SAD.

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&lt;big&gt;Melatonin, Bright Light Therapy and Dawn Simulation: Avery et al, 2001; Terman et al, 2001.&lt;/big&gt; 

[[w:Melatonin|Melatonin]] is a natural hormone which is used by all mammals to indicate that darkness is approaching and that it is time for rest. Research has suggested that the longer hours of darkness cause an imbalance to the melatonin rhythm and this is why people become more lethargic in the darker months (Brown, Pandi-Perumal, Trakht &amp; Cardinali, 2010).

Bright light therapy and dawn simulation both attempt to fix the melatonin imbalance and rhythm and restore proper sleep and mood ((Avery, Eder, Bolte, Hellekson, Dunner, Vitiell &amp; Prinz, 2001; Terman, Terman &amp; Cooper, 2001). To help reset the melatonin rhythm, bright light treatment is best administered in the morning eight and a half hours after melatonin onset(Terman et al., 2001). Bright light therapy has been criticised for the inconvenience caused to the patient.

Dawn simulation has been suggested as a better for the treatment of SAD (Avery et al, 2001). Dawn simulation is the use of dim lights slowly increasing in intensity until the patient wakes up and simulates the rising of the sun in the morning (Avery et al., 2001). Compliance of the patient is greater with dawn simulation than bright light therapy because it saves time and is very convenient (Avery et al., 2001).
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'''Use nature to improve your life''' 

Having knowledge of SAD can help you identify if this applies to you and what natural light treatments are available. Read &quot;Winter Blues: Everything You Need to Know to Beat Seasonal Affective Disorder&quot; by [[w:Norman_E._Rosenthal|Norman Rosenthal]] for more information. You should consult your doctor or psychologist for further information and diagnosis of SAD.

==Green exercise==
[[File:What is green exercise.svg|thumb|left|Table 1. ''Components of Green Exercise'']] 

Physical activity and exposure to nature both have positive effects on psychological well-being and physical health (Pretty, Peacock, Sellens &amp; Griffin, 2005). Table 1 shows the components that define [[w:Green_exercise|green exercise]]. Combining physical exercise and exposure to nature it termed green exercise and may promote even better physical and mental health than either one alone. [[Green_exercise]] incorporates environmental aesthetics and restorative environments. Engagement with nature can be seen on three levels (Mackay &amp; Neill, 2010): viewing nature, being in the presence of nature, and being involved in nature. Green exercise is about being involved in nature. Gardening, jogging outdoors, watersports and mountain biking can be considered as green exercise. 

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&lt;big&gt;Exercise and projected scenes: Pretty et al. 2005&lt;/big&gt;

{| border=&quot;1&quot; cellpadding=&quot;5&quot; cellspacing=&quot;0&quot; align=&quot;right&quot;
|-
! scope=&quot;col&quot; style=&quot;background:#9ACD32;&quot;|Condition
! scope=&quot;col&quot; style=&quot;background:#9ACD32;&quot;|Example
|-
!scope=&quot;col&quot; style=&quot;background:#F5DEB3;&quot; | Control
| No Image 
|-
!scope=&quot;col&quot; style=&quot;background:#F5DEB3;&quot; |Rural Pleasant
| Trees, fields etc 
|-
!scope=&quot;col&quot; style=&quot;background:#F5DEB3;&quot; |Urban Pleasant
| Cityscape with natural features ie city parks
|-
!scope=&quot;col&quot; style=&quot;background:#F5DEB3;&quot; |Rural Unpleasant
| Broken down cars in the bush etc
|-
!scope=&quot;col&quot; style=&quot;background:#F5DEB3;&quot; |Urban Unpleasant
| Broken windows, wire fencing etc 
|-
|}

There were five conditions with 20 participants. In four of the conditions 30 scenes were projected onto a wall whilst the participants exercised on a treadmill.  Blood pressure, self-esteem and mood were measured before and after the exercise. 

Results indicated that in the control group exercise reduced blood pressure and improved self-esteem and mood. When exercise was paired with pleasant scenes the mood and self-esteem of participants was higher than in other conditions. Unpleasant scenes when paired with exercise reduced the effect of exercise on self-esteem and self-esteem was lower in these conditions.
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'''Use nature to improve your life'''
[[File:Mountain bike ParcoSibillini.jpg|thumb|Green exercise]]

Next time you hop on your treadmill, consider facing it towards a window where you can see outside. Or go one step further and go for a jog outside, go hiking or go canoeing on the local lake. Being motivated to exercise can be hard, so check out the [[Motivation and emotion/Book/Exercise motivation|Exercise Motivation chapter]] for some tips.

== Miscellaneous tips on how to use nature to improve your life==

* Give flowers to a woman. When you do she will produce a [[w:Smile#Duchenne smiling|Duchenne smile]] and will be in a more positive mood for 3 days. (Haviland-Jones, Rosario, Wilson &amp; McGuire, 2005). 
* Make your backyard into a habitat for native flora and fauna. Build a bird bath and plant natives (requires less watering than non-natives and they look great).
* Go hiking and camping. Light a fire and star gaze at night (please check local fire bans beforehand).  
* Be in nature and be aware of your senses. Listen, smell, touch and watch (Louv, 2008). Try the [http://wilderdom.com/games/descriptions/SensualAwarenessInventory.html Sensual Awareness Inventory] for this activity.
* Plant a veggie garden. Good for saving money on produce. Great for the kids to see where their food comes from and be involved in the process (Louv, 2008).

==Summary==
Lessons about nature and psychological well-being can be learnt from the research and theories of environmental and evolutionary schools of psychology.  This chapter was written to help you realise that nature plays a vital role in our psychological well-being.  You now know that: 
* You can use restorative environments to reduce mental fatigue. (Attention Restoration Theory) 
* Being in touch with nature and our ancestral roots can improve your life. (Environmental Aesthetics &amp; Biophilia Hypothesis)
* Children need nature for their mental health. (Nature Deficit Disorder) 
* Emotions can be dramatically affected by the change of seasons. (Seasonal Affective Disorder)
* Combining exercise with nature is best. (Green exercise) 

The take away message from the chapter is this: '''go outdoors and use nature to improve your life'''.

==See also==
* [http://www.acfonline.org.au/news-media/acf-opinion/ten-reasons-why-we-need-more-contact-nature Ten reasons why we need more contact with nature] 
* [[Motivation and emotion/Book/Animals and emotion|Animals and emotion: How connection with animals can improve emotion and well-being]] 
* [[Motivation and emotion/Book/Weather and emotion | Weather and Emotion: How the weather effects our emotions]]
* [[/Transcript/|Transcript for the 5 minute audiovisual chapter overview]]

== References ==
{{Hanging indent|
Avery, D.H., Eder, D.N., Bolte, M.A., Hellekson, C.J, Dunner,D.L., Vitiello, M.V., &amp; Prinz, P.N. (2001). Dawn simulation and bright light in the treatment of sad: A controlled study. ''Society of Biological Psychiatry'', ''50'', 205-216.

Balling, J.D. &amp; Falk, J.H. (1982). Development of visual preference for natural environments. ''Environment and Behaviour'', ''14''(1), 5-28. 

Brown, G.M., Pandi-Perumal, S.R., Trakht &amp; Cardinali,. (2010). The role of melatonin in seasonal affective disorder. In T. Partonen, &amp; S.R Pandi-Perumal (Eds.), ''Seasonal Affective Disorder: Practice and research'' (pp. 149-162). New York, NY: Oxford University Press.

Buss, D.M. (2000). The evolution of happiness. ''American Psychologist'', ''55''(1), 15-23. 

Eagles, J.M. (2003). Seasonal affective disorder. ''The British Journal of Psychiatry'', ''182'', 174-176.

Fjeld, T. (1998a). Plants in interior surroundings—a source to health. ''gartneryrket 13''(15). 

Fjeld, T. (1998b). Plants; Light; Interior and health. ''research report. the norwegian radium hospital Olso''

Galindo, P. &amp; Rodriguez, J.A. (2000). Environmental aesthetics and psychological well-being: Relationships between preferences judgements for urban landscapes and other relevant affective responses. ''Psychology in Spain, 4'' (1), 13-27.

Gullone, E. (2000). The biophilia hypothesis and life in the 21st century: Increasing mental health or increasing pathology?. ''Journal of Happiness Studies, 1'', 293-321.

Haviland-Jones, J., Rosario, H.H., Wilson, P., &amp; McGuire, T.R. (2005). An environmental approach ot positive emotion: Flowers. ''Evolutionary Psychology, 3,''  104-132. 

James, W. (1892). ''Psychology: The briefer course''. New York: Holt.  

Kaplan, R. &amp; Kaplan, S. (1989). ''The experience of nature: A psychological perspective.'' Cambridge: Cambridge University Press. 

Khan, P.H. (1997). Developmental psychology and the biophilia hypothesis: Children’s affiliation with nature. ''Developmental Review, 17,'' 1-61.

Louv, R. (2008). ''Last Child in the Woods: Saving our children from nature-deficit disorder.'' Chapel Hill, NC: Algonoquin Books.

Mackay, G.J. &amp; Neill, J.T. (2010). The effect of green exercise on state anxiety and the role of exercise duration, intensity and greenness: a quasi-experimental study. ''Psychology of Sport and Exercise, 11,'' 238-245.

Nesse, R.M. (1989). Evolutionary explanations of emotions. ''Human Nature, 1'' (3). 261-289.

Pendse, B.P., Engstroem, G., &amp; Traeskman-Bendz, L. (2004) Psychopathology of seasonal affective disorder patients in comparison with major depression patients who have attempted suicide. ''Journal of Clinical Psychiatry 65'', 322–327.

Pretty, J.,  Peacock, J., Sellens, M., &amp; Griffin, M. (2005). The mental and physical health outcomes of green exercise. ''International Journal of Environmental Health Research, 15'' (5). 319-337.

Rosenthal, N. (2009). Issues for DSM-V: Seasonal affective disorder and seasonality. ''American Journal of Psychiatry, 166'' (8), 52-53.

Targum, S.D. &amp; Rosenthal, N. (2008). Research to practice; Seasonal affective disorder. ''Psychiatry'', May,31-33.

Terman, J., Terman, M., Lo, E., &amp; Cooper, T. (2001). Circadian time of morning light administration and therapeutic response in winter depression. ''Archives of General Psychiatry, 58,'' 69-75.

Ulrich, R.S. (1984). View through a window may influence recovery from surgery. ''Science, 224.'' 420-421.  

Wells, N.M. &amp; Evans, G.W. (2003). Nearby nature: A buffer of life stress among rural children. ''Environment and Behaviour, 35''(3), 311-330. 

Wilson, E. O. (1984). ''Biophilia''. Cambridge: Harvard University Press.}}

==External links==

* [http://www.thehappinessinstitute.com/blog/article.aspx?c=3&amp;a=1987 The Happiness Institute: Happiness and Nature]
* [http://ecco.vub.ac.be/?q=node/127 Evolutionary Well-Being: the paleolithic model]

[[Category:Motivation and emotion/Book/2011]]
[[Category:Motivation and emotion/Book/Nature]]
[[Category:Motivation and emotion/Book/Well-being]]</text>
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  <page>
    <title>Solar</title>
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      <text bytes="956" sha1="cuvczoxhm7e0v4y14e6l5qz3ivm35g3" xml:space="preserve">If you are looking for other projects, schools and content related to solar energy look here:

* [[Solar energy]]
* [[Introduction to solar energy]]

If you are looking for information on the solar system look for specific planets, for example:

* [[Solar System, technical/Mercury]]

==Idea==
Solar energy projects start with being able to aim solar light at a target and cook something like a frozen pizza. Then they branch out into hot water systems. After that the goal is to begin working with small solar cells. Working with lasers to power them and developing robots that can be powered by lasers. Typically these are robots climbing a rope and compete in a space elevator competition. The other expensive option is to design how adding solar cells to a [https://www.oneplacesolar.com university building].

==SubSystems==

==Tutorials==


==College/Universities==
:[[/Howard Community College/]]

[[Category:General_Engineering_Projects_2010-2012]]</text>
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      <comment>Reverted edit by [[Special:Contributions/Astitva1|Astitva1]] ([[User_talk:Astitva1|talk]]) to last version by [[User:IanVG|IanVG]] using [[Wikiversity:Rollback|rollback]]</comment>
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      <format>text/x-wiki</format>
      <text bytes="924" sha1="ax8md1ufmpghf0tglbchz4a2hfq35me" xml:space="preserve">If you are looking for other projects, schools and content related to solar energy look here:

* [[Solar energy]]
* [[Introduction to solar energy]]

If you are looking for information on the solar system look for specific planets, for example:

* [[Solar System, technical/Mercury]]

==Idea==
Solar energy projects start with being able to aim solar light at a target and cook something like a frozen pizza. Then they branch out into hot water systems. After that the goal is to begin working with small solar cells. Working with lasers to power them and developing robots that can be powered by lasers. Typically these are robots climbing a rope and compete in a space elevator competition. The other expensive option is to design how adding solar cells to a university building.

==SubSystems==

==Tutorials==


==College/Universities==
:[[/Howard Community College/]]

[[Category:General_Engineering_Projects_2010-2012]]</text>
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      <text bytes="27669" sha1="fbqmtpl47pa0ik6g4lybmirswkat6za" xml:space="preserve">{{title|Grief:&lt;br&gt;What it is and how to manage it?}}
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__TOC__

==Overview==
[[File:Rome WWStory angel in grief.jpg|150px|right|thumb|'''Figure 1'''. Caption goes here]]

{{RoundBoxTop|theme=4}}
{{quote|To spare oneself from grief at all cost can be achieved only at the price of total detachment, which excludes the ability to experience happiness&lt;br&gt;'''''- [[q:Erich Fromm|Erich Fromm]]'''''}}
{{RoundBoxBottom|theme=4}}

Grief is extremely stressful and can have a massive impact on an individual’s daily functioning. Grief is common and it is likely that you will experience grief if you haven’t already or grief will affect someone close to you. Knowing how to effectively deal with personal grief and the grief of others can be difficult. The present chapter aims to provide a comprehensive understanding of grief, how to effectively manage personal grief and the grief of others.

==What is grief?==
[[File:Senator Byrd funeral service.jpg|300px|thumb|right|Differences in mourning customs between cultures can be seen in funeral services]]

[[w:Grief|Grief]] is universal phenomenon that a can be thought of as any form of distress in response to loss (Howarth, 2011a). Grief is complex as it involves emotional, cognitive, physiological and behavioural manifestations. Grief is universal but its symptoms are not (Howarth, 2011a). People grieve for different lengths of time and in different ways as grief encompasses many emotions and its expression is unique to the individual (Howarth, 2011a). Grief effects people of all cultures and has been found to consistently effect people throughout history (Granek, 2010).

Often the words grief, bereavement and mourning are used interchangeably but they do have different meanings. Bereavement is grieving in response to the death of someone significant. Mourning refers to public displays of grief (Granek, 2010). Mourning is different cross culturally as it relates to cultural practices through which grief and bereavement are expressed (Stroebe, Hansson, Schut, &amp; Stroebe, 2008). Mourning is related to the customs of a society as it is influenced by social context. Differences in mourning customs between cultures can be seen in religious funeral services (Stroebe et al., 2008).

==Theories of grief==
{{expand}}

===Bowlby===

[[File:Toposa mother and child.jpg|175px|thumb|left|According to Bowlby people form attachments for protection and survival]]

John Bowlby is most recognized for his work on attachment theory and has used attachment theory to provide a biological explanation of grief (Stroebe, 2002). Attachment can be defined as a meaningful bond that is made with a significant something or someone and is important for maintaining relationships (Field, 2006). Bowlby thought that people formed attachments for protection and survival. An attachment relationship that serves a protection and survival function can be seen in the attachments made between mother and child (Mikulincer &amp; Shaver, 2008). 

Bowlby theorised that when someone feels threatened or distressed their attachment behavioural system is activated and they will seek protection and comfort from those who they have formed an attachment with. When a significant attachment figure dies or is lost the individual will experience extreme distress which Bowlby called separation distress (Mikulincer &amp; Shaver, 2008). The individual fears life without their attachment figure as they can no longer receive support from them.  The individual will experience grief until they become detached from their former attachment. According to Bowlby reorganisation happens when the individual accepts the loss and integrates the loss into their future (Mikulincer &amp; Shaver, 2008). This model of grief has been criticised as not all individuals grieving styles follow distinct stages however stages are useful in informing people what is involved in the grieving process (Mikulincer &amp; Shaver, 2008).

===Kübler-Ross===
Swiss psychiatrist Elizabeth Kübler-Ross proposed a stage theory of grief from her work with terminally ill hospital patients (Krueger, 2006). Kübler-Ross’ theory of grief was used to explain the process of grief in terminally ill patients and their loved ones reactions to loss (Krueger, 2006). Kübler-Ross’ theory consists of five stages: denial, anger, bargaining, depression and acceptance. The theory has been criticised as it comes from Kübler-Ross’ Subjective experiences with dying people and the theory does not take into account individual differences (Retsinas, 1988).   
{{Robelbox|theme={{{theme|1}}}|title=Kübler-Ross' Stage Theory of Grief}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;  

'''Denial''' – when people first face loss they enter a state of denial which may last for months. In this stage of Kübler-Ross’ theory people are in shock and find it easier to deny the reality that they have a terminal illness (Bolden, 2007).

'''Anger''' – after denial comes anger. The grieving person becomes extremely angry about anything and everything as they may feel it is unfair that they have been singled out. this stage can be difficult for carers or loved ones of the grieving person as they can become a target of the anger (Bolden, 2007).

'''Bargaining''' – once the grieving person has passed the anger stage they enter a bargaining stage. Within this stage the individual realises that death is inevitable but pleads for more time and feels that they would do anything for more time. The individual bargains with doctors, their god or just with themselves. It is within this stage that any unfinished business or loose ends are dealt with (Bolden, 2007).

'''Depression''' – the individual becomes depressed after they have taken care of unfinished business as they are starting to accept the inedibility of the loss. it is thought that the individual will experience reactive depression which refers to grieving losses that have already been experienced. After reactive depression comes proactive depression which is the grieving of future losses (Bolden, 2007). 

'''Acceptance''' – the last stage of Kübler-Ross’ theory is acceptance. The individual becomes detached and accepts the inevitability of death (Bolden, 2007).
  &lt;/div&gt;
  {{Robelbox/close}}

==Trajectories of grief==
A study was conducted by Bonanno et al. (2002) that aimed to identify the most common trajectories of grieving. Bonanno et al. used data from previous studies that had interviewed widows before and after their partners had died. Bonanno et al (2002) found five trajectories of grief.

{{Robelbox|theme={{{theme|2}}}|title=Trajectories of Grief}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Common grief ''' – these people have low preloss depression, high postloss depression 6 months after loss but recover at 18 months after the loss (Bonanno et al., 2002).

'''Stable low distress ''' – these people have low preloss depression and low postloss depression at 6 and 18 months after the loss (Bonanno et al., 2002).

'''Depression then improvement ''' – these people have high preloss depression but low postloss depression at 6 and 18 months after the loss (Bonnano et al., 2002).

'''Chronic grief ''' – these people showed low preloss depression and high postloss depression at 6 and 18 months after the loss (Bonanno et al., 2002)

'''Chronic depression''' – these people have high preloss depression and high postloss depression 6 and 18 months after the loss (Bonanno et al., 2002).

  &lt;/div&gt;
  {{Robelbox/close}}

==Physiology and grief==
[[File:Heart-beat.gif|300px|thumb|right|Grief affects cardiovascular health]]

Research by Kowalski and Bondmass (2008) have found that grief causes significant physiological change. Participants of the study were 173 widowed women who completed questionnaires relating to their grief. The women were asked if they had experienced any physical symptoms after the death of their partner. Results of the study found that the women reported having experienced pain, trouble sleeping, symptoms that required surgery, gastro intestinal issues and neural and circulatory problems (Kowalski &amp; Bondmass, 2008). A similar study by Fogel (2007) found that grief and depression affect cardiovascular health.

==Types of grief==
{{expand}}

===Normal grief===
[[File:A child sad that his hot dog fell on the ground.jpg|200px|thumb|left]]

Grief involves complex emotions that affect people differently. Grief is subject to cultural variation and at times can be difficult to distinguish from Major Depressive Disorder (Howarth, 2011a).  This makes it hard to define what normal grief is, but a defining feature of normal grief is reconciliation (Doughty, 2009). Reconciliation is achieved when an individual no longer grieves a loss and can move on with their life. To achieve reconciliation, it is thought that people must accept the loss, fully grieve the loss, make adjustments to their life, incorporate the loss into their self-concept, acknowledge that what was lost will not be a part of their future but rather a memory, find meaning from the loss to be able to move on and make new bonds in life (Doughty, 2009). 

As grief affects people differently research suggests that there may be different styles of grieving that are shaped by personality, gender and culture (Doughty, 2009). Research proposes that individuals have adaptive grieving styles that influence their behaviour, cognitions and strategies used to deal with loss. Adaptive grieving styles are distinguished by how an individual outwardly expresses a loss and how a loss is experienced internally (Doughty, 2009). According to the adapting grieving styles theory, an individual’s grieving style can range between intuitive and instrumental grieving (Doka, 2002). 

Intuitive grief refers to a greater expression of emotion as a result of the loss and a desire to talk about the loss. Intuitive griever’s outward expressions reflect their inner feelings (Doughty, 2009).  Intuitive grievers feel sad and express their sadness through crying and talking about their loss. Intuitive grief is stereotyped as a female grieving style (Doka, 2002). Instrumental grievers do not display their emotions as intuitive grievers do because they do not feel the need to (Doughty, 2009). This may be because their grief is not experienced as intensely as an intuitive griever experiences grief. Instrumental grievers aim to control their emotions and may see their grief as a problem solving challenge (Doughty, 2009). People with an instrumental grieving style appear unaffected by loss as they do not express grieving related behaviours such as crying (Doughty, 2009). Instrumental grief is stereotyped as a male grieving style (Doka, 2002). The most common adaptive grieving style is a blended style that incorporates both intuitive and instrumental grief (Doughty, 2009).

===Complicated grief===
Complicated grief is experienced by people who are unable to overcome grief (Strada, 2011). Their grief response is considered abnormal as it is persistent in its intensity and duration. It is thought that complicated grief arises as a result of deficits in emotional regulation (Gupta &amp; Bonanno, 2011). The symptoms of complicated grief are similar to Major Depressive Disorder symptoms but can be distinguished by an individual’s extreme longing for what is lost, strong disbelief of the loss and feelings of loneliness and detachment (Strada, 2011). People with complicated grief also feel that their life no longer has meaning after loss (Strada, 2011). As complicated grief is different from Major Depressive Disorder it will be included in the next edition of the Diagnostic and  Statistical Manual of mental Disorders as Prolonged Grief (Strada, 2011).

===Disenfranchised grief===
[[File:So happy smiling cat.jpg|150px|thumb|right| The death of a pet is not a socially acknowledged loss]]
Disenfranchised grief is when a person is denied their right to grieve. The individual still experiences grief but it is not socially acknowledged or validated (Doka, 2002). The grief is not acknowledged for many reasons, such as the nature of the relationship to what is lost is not socially acceptable, or the manner in which the individual mourns is not socially acceptable (Doka, 2002). Loss that is not socially acknowledged can include the death of a pet, the death of a unborn  baby, a missing family member or friend, losing a job, having a family member on death row or losing property in a fire (Attig, 2004). When an individual’s grief is disenfranchised, society denies them a grieving role which limits their support networks and they are denied compensations such as time off from work (Doka, 2002). 

It is important to understand that people can become attached to many things and any form of loss can cause someone to grieve. To help someone recover from loss, their grief needs to be recognised and acknowledged (Attig, 2004). Being empathetic and respectful towards someone’s loss is ethical and prevents their grief from becoming disenfranchised (Attig, 2004). 

 {{Robelbox|theme={{{theme|5}}}|title=Grief is disenfranchised when}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''The relationship is not recognised''' – most societies consider family relationships to be the most important. Friendships and attachments to material possessions are considered less important (Doka, 2002).

'''The loss is not acknowledged''' – societal rules decide what losses are significant and what losses are not (Doka, 2002).   

'''The griever is excluded''' – sometimes people are socially defined as incapable of grief. This can be seen in situations where a person has a disability or mental illness and is excluded from grieving rituals such as planning funeral services because their grief is not acknowledged (Doka, 2002). 

'''Circumstances of the death''' – the way in which a loved one dies can disenfranchise grief. People who lose a significant person in their life to AIDS, homicide or suicide feel stigma which prevents them from receiving full support (Doka, 2002). 

'''The way an individual grieves''' – grief effects people differently. Support is more likely to be given to people who have strong immediate reactions in response to loss than people who grieve later (Doka, 2002).  

  &lt;/div&gt;
  {{Robelbox/close}}

===Anticipatory grief===
Anticipatory grief is grief that results from the knowledge of an impending loss (Cheng, Lo, Chan, Kwan &amp; Woo, 2010). Grieving prior to a loss can be experienced by the carers of loved ones with a terminal illness or people who have a terminal illness (Cheng et al., 2010). Anticipatory grief refers to any loss that is anticipated, not only to anticipating death from terminal illness. Yet most of the research available on anticipatory grief involves people who are terminally ill and the effect anticipatory grief has on their families (Cheng et al., 2010). 

Research has found that terminally ill cancer patients deal with many losses that are associated with anticipatory grief when they learn that they only have a certain time left to live (Cheng et al., 2010). Their losses include the loss of a future with friends and family, loss of hopes and dreams for the future, loss of cognitive functioning, loss of identity and role definition and the loss of independence as their cancer worsens (Cheng et al., 2010). Carers of people with terminal illnesses experience anticipatory grief as they fear the eventual death of their loved one but also experience loss related to the kind of illness their loved one has died from (Holley &amp; Mast, 2010). Carers of people with dementia grieve the loss of the relationship that they had with the dementia sufferer as dementia symptoms worsen and affect cognitive and physical abilities (Holley &amp; Mast, 2010). Caregiver’s anticipatory grief is also associated with the sacrifices, burdens and isolation that come with caring for someone with a terminal illness (Al-Gamal &amp; Long, 2010).

==Grief in childhood and adolescence==

Children experience grief but experience it differently to adults. Depending on the age of the child they may not understand that death is forever (Willis, 2002). As children develop they learn that some things are lost forever, can grieve previous losses that were not originally understood (Willis, 2002). Children grieve in bursts that are expressed behaviourally and emotionally. For children to effectively recover from loss they need a safe and stable environment and an adult who they can depend on (Himebauch, Arnold &amp; May, 2008). 
{{Robelbox|theme={{{theme|7}}}|title=Age and Grief}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;  

'''0-2 years''' – children under 2 years of age do not understand death and loss but do experience grief in situations such as when they are separated from their parents or carers (Himebauch et al., 2008). in accordance with Bowlby's attachment theory, when infants are separated from their parents or carers they will experience separation distress.

'''2-6 years''' – preschool aged children do not understand the permanency of death and loss. They do feel grief when they lose something close to them and will often feel that the loss was their fault. It is important to let the child know that they did not cause the loss and provide them with factual explanations (Himebauch et al., 2008).

'''6-8 years''' – primary school aged children know that death is permanent. They experience grief and may fear death. To help the child cope with their grief school teachers should be informed of the loss (Himebauch et al., 2008).

'''8-12 years''' – teenagers understand death as adults do and grieve in much the same way but may have trouble expressing their feelings. It is not uncommon for teenager to engage in risky behaviour to challenge their mortality. Grieving teenagers need freedom to spend time with their friends and supportive environment (Himebauch et al., 2008).
  &lt;/div&gt;
  {{Robelbox/close}}

==Managing and overcoming grief==
[[File:Light has gone out.jpg|250px|thumb|left|Writing about loss has been suggested as a way to manage grief]]
Grief counselling is a method used to alleviate peoples grieving and aims to help people grow from the traumatic experiences caused by grief (Altmaier, 2011). Most people will overcome their grief in time but for those whose grief appears to be chronic or for people with anticipatory grief, grief counselling is a useful treatment. It is thought that grief counselling creates a sense of hope for the future in the grieving client (Cutcliffe, 2006). By experiencing a caring human connection with the counsellor the client is able participate in a positive relationship. The counselling relationship is a stable relationship that provides the client with consistent support (Cutcliffe, 2006).

Another approach to managing grief could be writing about it. Research by Lichtenthal and Cruess (2010) studied the effect that writing about loss has on grief. The grieving participants in this study wrote for 20 minutes about their grief. Their writing focused on trying to make sense of their grief and trying to identify the good aspects of their life (Lichtenal &amp; Cruess, 2010). The results found that spending time writing about the good aspects of their life after loss and trying to make sense of personal grief was effective in helping people adjust to life after loss (Lichtenal &amp; Cruess, 2010).

For children to overcome grief they must be able to accept that what they have lost is gone forever. For this to happen they need to be informed of the facts surrounding the loss (Howarth, 2011b).  Children need to be given time to grieve the loss and experience the pain that is associated with the knowledge that the loss is permanent. After children have grieved the loss the feelings of grief becomes less frequent and they can learn to how to live without what was lost and convert the loss into a memory rather than something that remains present (Howarth, 2011b). Specific factors that influence how children overcome and manage grief is the child’s age, the safety of their environment and stability and support of their caregivers. Children manage their grief better when they are supported emotionally by their caregivers and when they feel safe in their environment. If children grieve as a result of the death of someone close to them, they should be allowed to participate in rituals such as funeral services to help them overcome their grief (Howarth, 2011b).

==Conclusion==
*Grief is any distress in response to loss.
*Grief is universal and effects people differently. 
*The five most common trajectories of grief are: Common grief, Stable low distress, Depression then improvement, chronic grief and Chronic depression.
*It is thought that that people grieve because they experience separation distress when they can no longer make contact with what is lost. 
*When faced with loss people will pass through the stages of denial, anger, bargaining, depression and acceptance. 
*Grief causes significant physiological change and has been found to affect physiological health.
*Different types of grief include; normal grief, complicated grief, disenfranchised grief and anticipatory grief. 
*Normal grief involves reconciliation and recovery from grief.
*Complicated grief is experienced by people who are unable to overcome grief.
*Disenfranchised grief is when a person is denied their right to grieve.
*Anticipatory grief results from the knowledge of an impending loss.
*Children grieve differently to adults. 2 year old do not understand loss, 2-6 year olds do not understand the permanency of death and loss, 6-8 year olds know that death is permanent and 8-12 year olds understand the permanency of death and grieve in much the same was as adults do. 
*Ways to manage and overcome grief include grief counselling and writing about loss.
*Grieving children need a safe environment and support from their caregivers to be able to overcome grief effectively.

== Quiz ==
&lt;quiz display=simple&gt;
{The stage theory of grief that involves the stages: denial, anger, bargaining, depression and acceptance was proposed by?
|type=&quot;()&quot;}
- Bowlby.
- Bonanno.
+ Kübler-Ross.


{Anticipatory grief is grief that results from the knowledge of an impending loss.
|type=&quot;()&quot;}
+ True.
- False.


{People grieve for different lengths of time and in different ways.
|type=&quot;()&quot;}
+ True.
- False.

{preshcool aged children understand death as permanent.
|type=&quot;()&quot;}
- True.
+ False.
&lt;/quiz&gt;

==See also==
* [[w:Grief|Grief]] (Wikipedia)
* [[w:Kubler-Ross model|Kubler-Ross model]] (Wikipedia)
* [[w:John Bowlby|John Bowlby]] (Wikipedia)
* [[w:George Bonanno|George Bonanno]] (Wikipedia)
* [[w:Grief Counseling|Grief counseling]] (Wikipedia)

==References==
&lt;!-- Put references between the divs and they'll be hanging indented --&gt;
&lt;div style=&quot;padding-left: 2em; text-indent: -2em&quot;&gt;
Al-Gamal, E., &amp; Long, T. (2010). Anticipatory grieving amoung parents living with a child with cancer. ''Journal of Advanced Nursing, 66,'' 1980-1990.

Altmaier, E. M. (2011). Best practices in counseling grief and loss: Finding benefit from trauma. ''Journal of Mental Health Counseling, 33,'' 33-45.

Attig, T. (2004). Disenfranchised grief revisited: Discounting hope and love. ''Journal of Death and Dying, 49,'' 197-215. 

Bolden, L. A. (2007). A review of on grief and grieving: Finding the meaning of Grief through the five stages of loss. ''Counselling and Values, 51,'' 235-237.

Bonnano, G. A., Wortman, C. B., Lehman, D. R., Tweed, R. G., Haring, M., Sonnega, J., et al. (2002). Resilience to loss and chronic grief: A prospective study from pre-loss to 18 months post-loss. ''Journal of Personality and Social Psychology, 83,'' 1150-1164.

Cheng, J. O. Y., Lo, R. S. K., Chan, F. M. Y., Kwan, B. H. F., &amp; Woo, J. (2010). An exploration of anticipatory grief in advanced cancer patients. ''Psycho-Oncology, 19,'' 693-700.

Cutcliffe, J. R. (2006). The principles and processes of inspiring hope in bereavement counselling: A modified grounded theory study—part one. ''Journal of Psychiatric and Mental Health Nursing, 13,'' 598-603. 

Doka, K. J. (2002). Introduction. In K. J. Doka (Ed.), ''Disenfranchised grief: New directions, challenges and strategies for practice'' (pp. 5-22). Champaign, IL: Research Press.

Doughty, E. A. (2009). Investigating adaptive grieving styles: A Delphi study. ''Death Studies, 33,'' 462-480.

Field, N. P. (2006). Unresolved grief and continuing bonds: An attachment perspective. ''Death studies, 30,'' 739-756.

Fogel, J. (2007) Negative affect and cardiovascular health. ''Journal of Cognitive and Behavioural Psychotherapies, 7,'' 107-113.

Granek, L. (2010). Grief as Pathology: The evolution of grief theory in psychology from Freud to the present. ''History of Psychology, 13,'' 46-73. 

Gupta, S., &amp; Bonanno, G. A. (2011). Complicated grief and deficits in emotional expressive flexibility. ''Journal of Abnormal Psychology, 120,'' 635-647.

Himebauch, A., Arnold, R. M., &amp; May, C. (2008). Grief in children and developmental concepts of death #138. ''Journal of Palliative Medicine, 11,'' 242-243.

Holley, C. K., &amp; Mast, B. T. (2010). Predictors of anticipatory grief in dementia caregivers. ''Clinical Gerontologist: The Journal of Aging and mental Health, 33,'' 223-236.

Howarth, R. A. (2011a). Concepts and controversies in grief and loss. ''Journal of Mental Health Counselling, 33,'' 4-10.

Howarth, R. A. (2011b). Promoting the adjustment of parentally bereaved children. ''Journal of Mental Health Counseling, 33,'' 21-32.

Kowalski, S. D., Bondmass, M. D. (2008). Physiological and Psychological symptoms of grief in widows. ''Research in Nursing and Health, 31,'' 23-30.

Krueger, G. (2006). Meaning-Making in the aftermath of sudden infant death syndrome. ''Nursing Inquiry, 13,'' 163-171.

Lichtenthal, W. G., &amp; Cruess, D. G. (2010). Effect of directed written disclosure on grief and distress symptoms among bereaved individuals. ''Death Studies, 34,'' 475-499.

Mikulincer, M., &amp; Shaver, P. R. (2008) An attachment perspective on bereavement. In M. S. Stroebe, R. O. Hansson, H. Schut, &amp; W. Stroebe (Eds.), ''Handbook of bereavement research and practice'' (pp. 87-112). Washington, DC: American Psychological Association. 

Retsinas, J. (1988). A theoretical reassessment of the applicability of Kübler-Ross’s stages of dying. ''Death Studies, 12,'' 207-216.

Strada, E. A. (2011). Complicated Grief. In S. H. Qualls, &amp; J. E. Kasl-Godley (Eds.), ''End of life issues, grief, and bereavement: What clinicians need to know'' (pp. 181-200). Hoboken, NJ: Wiley and Sons. 

Strobe, M. S. (2002). Paving the way: From early attachment theory to contemporary bereavement research. ''Mortality, 7,'' 127-138.

Stroebe, M. S., Hansson, R. O., Schut, H., &amp; Stroebe, W. (2008) Bereavement research: Contemporary perspectives. In M. S. Stroebe, R. O. Hansson, H. Schut, &amp; W. Stroebe (Eds.), ''Handbook of bereavement research and practice'' (pp. 3-25). Washington, DC: American Psychological Association. 

Willis, C. A. (2002). The grieving process in children: Strategies for understanding and reconciling childrens perceptions of death. ''Early Childhood Education Journal, 29,'' 221-226.
&lt;/div&gt;

==External links==
* [http://www.lifeline.org.au Lifeline]
* [http://www.kidshelp.com.au Kids Helpline]
* [http://helpguide.org/mental/grief_loss.htm Coping with Grief and Loss]
* [http://www.youthbeyondblue.com/factsheets-and-info/fact-sheet-16-dealing-with-loss-and-grief/ Youth Beyondblue]

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
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      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
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== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.2A.CLA.20260610.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260610.pdf|B]] ||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

== Barrel Shifter ==
* Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]])
&lt;/br&gt;

'''Mux Based Barrel Shifter'''
* Analysis  ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) 
* Implementation 
&lt;/br&gt;

== Multiplier == 
=== Array Multipliers ===
* Analysis  ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]])
&lt;/br&gt;

=== Tree Mulltipliers ===
* Lattice Multiplication  ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]])
* Wallace Tree  ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]])
* Dadda Tree  ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]])
&lt;/br&gt;

=== Booth Multipliers ===
* [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]]
* Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]])
&lt;/br&gt;

== Divider == 
* Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])&lt;/br&gt;

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

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      <text bytes="2050" sha1="6i8rhjvowdownaprvm7nenv5gnvw884" xml:space="preserve">{{RoundBoxTop|theme=14}}{{title|Motivation and emotion (Book)}}&lt;div align=&quot;center&quot;&gt;&lt;small&gt;Understanding and improving our motivational and emotional lives using psychological science&lt;/small&gt;&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;&lt;small&gt;''Use this page to navigate over 1,800 chapters''&lt;/small&gt;&lt;/div&gt;{{RoundBoxBottom}}__NOTOC__
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Browse the chapters by year, topic, or enter a search term above. Or continue reading [[/Overview|about this project ...]]

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==Overview==
Are you, a significant other or a loved one unemployed? Is unemployment having an effect on your [[Motivation|motivation]] to search for a job? People are considered unemployed if they are &quot;currently without work, but actively seeking [[Wikipedia:Unemployment|employment&quot;.]] Unemployment is becoming an increasing reality to many individuals with the unemployment rates increasing in Australia in just one {{what}}season by 11000 people (Australian Bureau of Statistics, 2014). It can have detrimental effects on motivation, pushing an individual to search for a job for a number of reasons. This chapter discusses unemployment and its effect on job search behaviour, or work motivation. Using theories including the Self-Determination Theory, Learned Helplessness Theory and the Expectancy-Value Theory, it will help to understand the underlying factors as to how and why job search behaviour may be affected by lack of employment. Finally, it will look at how the effects of unemployment on motivation and job search behaviour can be minimised and improved.

'''Work motivation''' “is a set of energetic forces that originate both within as well as beyond an individual’s being, to initiate work-related behaviour, and to determine its form, direction, intensity, and duration.” (Pinder, C. C., 1984, p11).

'''Job search behaviour;''' &quot;the product of a dynamic self-regulatory process that begins with the identification of and commitment to an employment goal. This goal subsequently activates search behavior designed to bring about the desired goal&quot; (Kanfer et al., 2001, as cited in Klehe &amp; van Hooft, 2001, p4)

==Consequences of unemployment==
Studies show that unemployment can have detrimental effects on the psychological well-being of the unemployed individual{{fact}}. Johada (1982) proposed the Latent Deprivation Model, theorising that employment provides gains in several psychological aspects, therefore unemployment supports the lack of these gains, often causing poor psychological well-being (Creed &amp; Macintyre, n.d.). Johada’s (1982) theory and train of thought in general, in particular those of the psychological impacts of unemployment, have been proven via many studies{{fact}}.


A meta-analysis conducted (Murphy &amp; Athanasou, 1999) found that unemployment has a negative impact on psychological well-being{{rewrite}}{{expand}}. 
A study by Kuhnert (1989) followed Johada’s (1982) theory and has suggested that, to better understand the consequences of unemployment, focus needs to be first put on the gains individuals achieve whilst employed. This study suggests that financial gains, social contact and inclusion, well-being and a sense of belonging are included in the gains of employment and suggest that when unemployed these gains are lost causing feelings of meaninglessness (Kuhnert, 1989).

The first consequence of unemployment that comes to mind is financial strain, however, in addition to this there have been studies carried out to find many psychological effects of unemployment. Many studies have found that due to sudden financial strain and the lack of structure and satisfaction previous employment may have offered, individuals may suffer with mental health issues such as [[Wikipedia:Depression (mood)|depression]] (Vansteenkiste, Lens, De Witte &amp; Feather, 2005; Vansteenkiste, Lens, De Witte, De Witte &amp; Deci, 2004) and [[Wikipedia:Anxiety|anxiety]] (Vansteenkiste et al., 2004), as well as somatic symptoms (Vansteenkiste et al., 2004) such as feelings of meaningless (Vansteenkiste et al., 2005), lack of life satisfaction and general feelings of unhappiness (Vansteenkiste et al., 2005). In addition, Vansteenkiste et al. (2004 &amp; 2005) stated that other studies have suggested higher rates of [[Wikipedia:Suicide|suicide]] occur in unemployed individuals.

Findings from the Stress and Well-being in Australia [http://www.psychology.org.au/Assets/Files/Stress%20and%20wellbeing%20in%20Australia%20survey%202013.pdf survey] (Australian Psychological Society, 2013) have shown that unemployed individuals in Australia show significantly lower levels of well-being and are the higher reporters of anxiety and depressive symptoms in comparison with employed people. ''Figure 3'' shows that unemployed individuals are more likely to incur a [[Wikipedia:Mental disorder|mental disorder]] than any other employment status group.

[[File:Bar Graph. Mental Disorders in the work force. Australia.gif|200px|framed|center|''Figure 3''. Comparison of employment status and percentage of mental disorders in Australia, 2007 (ABS, 2008).]]

Although a majority of cases of unemployment on individuals results in a negative impact on psychological well-being (Murphy et al., 1999), unemployment occasionally effects individuals in a positive manner in producing the feeling of being free (Vansteenkiste et al., 2004). However, this usually only occurs when the individual's employment prior to becoming unemployed caused feelings of dissatisfaction (Vansteenkiste et al., 2004).

==Theories==
There are many theories which can be used to understand the effect unemployment has on the motivation an individual experiences when engaging in job search behaviour. Three theories in put the effects of unemployment on motivation into context: Deci and Ryan’s (1985) Self-Determination Theory, Seligman’s (1975) Learned Helplessness Theory, and Bandura's (1977) Expectancy-Value Theory.

====Learned helplessness theory====
Seligman (1975) proposed the [[Wikipedia:Learned helplessness|learned helplessness theory]] which suggests an individual experiences 'learned helplessness' because of past failings and it occurs when an individual perceives outcomes as uncontrollable (Rodriguez, 1997). The theory suggests that if an individual has previously failed at a task, for example; to gain a job after several applications, the individual will assume helplessness because of believed incapability, and discontinue attempts of achieving their original goal. According to Seligman (1975), this perception on the individual's part causes learned helplessness, which  produces cognitive, motivational and emotional deficits. These three deficits are also seen in individuals living with depression (Rodriguez, 1997). In the context of unemployment, this theory seems to have high value as whilst in a situation of unemployment it is likely that helplessness and uncontrollability is experienced (Rodriguez, 1997).

====Self-determination theory====
[[Wikipedia:Self-determination theory|Self-determination theory]] (SDT) focuses on types of motivation. In line with SDT, these types of motivation are initially categorised into 1) [[Motivation and emotion/Book/2013/Intrinsic motivation|Intrinsic Motivation]] and 2) [[Motivation and emotion/Book/2013/Extrinsic motivation|Extrinsic Motivation]]. SDT proposes that &quot;different types of motivation will result in different types of outcomes&quot; (Vansteenkiste et al., 2005. p272). Along with the two types of motivation categorised as intrinsic and extrinsic, SDT discusses two additional sub-categories of which are named autonomous and controlled motivation. Controlled motivation seems to have just the one type which can be defined as being pressured to engage in a behaviour (Deci, 2012). Ryan and Deci (2000) believe there are two types of autonomous motivation; intrinsic and identified. These are described as &quot;two flavours&quot; by Deci (2012) and consist of &quot;interest and enjoyment&quot; and &quot;values and beliefs&quot; to engage in a behaviour.

*'''Intrinsic Motivation;''' &quot;doing something because it is inherently interesting or enjoyable.&quot; (Deci &amp; Ryan, 2000, p55)
*'''Extrinsic Motivation;''' &quot;doing something because it leads to a separable outcome.&quot; (Deci &amp; Ryan, 2000, p55)


{{RoundBoxTop|theme=2}}
Sarah is unemployed. She spends her days searching for jobs because she genuinely enjoys the prospect of what new jobs there will be available to apply for each day, imagining herself in the role while applying. Sarah is displaying the use of intrinsic motivation.
{{RoundBoxBottom}}
{{RoundBoxTop|theme=3}}
John is unemployed. He spends his days searching for jobs online because he has to prove to centrelink that he has applied for 20 jobs a week in order to receive his benefits. John is displaying the use of extrinsic motivation.
{{RoundBoxBottom}}

[[File:SelfDeterminationTheory.png|300px|framed|''Figure 4''. Self-determination theory and the three psychological needs; Competence, Relatedness and Autonomy]]

SDT theorises that an individual must feel three psychological needs (autonomy, competence and relatedness) (Vansteenkiste et al., 2005) in order to gain the highest levels of motivation. 
*'''Competence''' refers to an individual having a sense of task mastery and the feeling of improving their skills and knowledge. 
*'''Relatedness''' refers to the sense of belonging
*'''Autonomy''' refers to the sense of control in terms of their personal goals and behaviour
(Vansteenkiste et al., 2005). 

Deci and Ryan (1985) theorised that if these three psychological needs are met, the individual becomes intrinsically self-determined to pursue their goal (as cited in Vansteenkiste et al., 2005).

====Expectancy-value theory====
[[Wikipedia:Expectancy-value theory|The expectancy-value theory]] (EVT) is cognitive-motivational theory and, as the name suggests, there are two aspects; expectancies and values. Bandura (1977) theorised that the motivation levels in an individual to achieve a goal is associated with the individual's expectation of achieving the goal, along with the value of which the individual has put on the goal being achieved (Vansteenkiste et al., 2005). In line with EVT, if the individual assesses a task, such as looking for a job, as having personal value as well as the expectation of success, the individual is likely to attempt the task.

Within this theory, Bandura (1977) suggests there are two types of expectancies; efficacy-expectations and outcome expectations:
*'''Efficacy-expectations''' - &quot;the conviction that one can successfully execute the required behaviour to produce the outcome.&quot; (Bandura, 1977, as cited in Vansteenkiste et al., 2005). This definition can be interpreted as the individual's expectations surrounding their ability to actually produce the behaviour that can in turn, create the wanted outcome.
*'''Outcome expectations''' - &quot;a person's estimate that a given behaviour will lead to certain outcomes.&quot; (Bandura, 1977, as cited in Vansteenkiste et al., 2005). 

EVT suggests that along with expectancies, values play a major role on an individual's motivation levels. If an individual places high value on an outcome, such as getting and job and therefore becoming employed, the more likely the individual is to have high levels of motivation, and in this case, high levels of job search behaviour. 

==What does the literature say?==
There have been many studies conducted surrounding the issues of the effect unemployment has on motivation and therefore, job search behaviour as well as many journal articles compiled on this same topic. Many of these base their research upon a theory or two, providing a comparative report. In the following section some of these are discussed.

====A look at self determination theory in context====
Vansteenkiste et al., (2004) conducted two studies using the theoretical basis of SDT, to find out the ‘why’ and ‘why not’ of the motivation behind job search behaviours in individuals. The first study conducted involved 254 unemployed individuals and was used to produce an accurate questionnaire that would be able to predict autonomy, control and amotivation in terms of job search behaviour and the lack of motivation associated with not searching. (Vansteenkiste et al., 2004). The second study conducted was used to analyse the relationships between job search behaviour, experiences of unemployment and well-being of an unemployed individual (Vansteenkiste et al, 2004). In line with SDT the hypotheses of these studies were that concepts associated with levels of autonomy, such as autonomous and controlled motivation were probable of having high correlation (Vansteenkiste et al., 2004). Furthermore, it was hypothesised that concepts such as autonomous motivation and amotivation would show less correlation than those concepts mentioned in the first hypothesised statement (Vansteenkiste et al, 2004). The results showed that the hypothesised statements were correct, showing that autonomous motivation, in this case to engage in job search behaviour, was positively correlated with concepts that involve a level of autonomy (Vansteenkiste et al., 2004). On the other hand, amotivation was negatively linked to autonomous motivation (Vansteenkiste et al., 2004). The results directly align with the SDT where it is theorised that autonomous motivation is required in order to gain the most motivation to acquire a personal goal, such as gaining employment.

[[File:Digital-student.jpg|200px|thumbnail|''Figure 5''. Autonomous motivation positively effects job search behaviour{{fact}}]]

Gagne &amp; Deci (2005) wrote an article on SDT and its relationship with work motivation. The purpose of the article was to explain how SDT best suits the topic of work motivation (Gagne et al., 2005). SDT explains the intrinsic and extrinsic motivation in a clear and concise way in order to fully understand the way work motivation occurs, making it a more clear and concise theory to use in this domain. SDT aims to prove that where both intrinsic and extrinsic motivators are present, work motivation is high. However, Gagne et al. (2005) discuss that early studies conducted inferred that in work situation where extrinsic motivators were highly present in comparison to intrinsic motivators, work motivation could be at a lower level. Therefore, when understanding the effects of intrinsic and extrinsic motivators on work motivation and even job search behaviour, it is clear that intrinsic motivators are most important in the scenario in order to gain high levels of motivation whether searching for a job, or currently employed. In addition to SDT creating an understanding of both intrinsic and extrinsic motivators and their effects on motivation, it also offers an understanding of the effects extrinsic motivators have on intrinsic motivators (Gagne et al., 2005). As discussed above, SDT discusses an individual’s needs of autonomy, competence and relatedness. Gagne et al. (2005) state that there have been many studies including Reis et al (2000) of which tested these needs in an individual for use in everyday life. Reis et al (2000) found that the well-being of an individual altered in line with the level of satisfaction of these three needs (as cited in Gagne et al., 2005). This research can be utilised in the domain of work motivation and job search behaviour as it assists in the understanding surrounding personal well-being at any given time, therefore providing understanding as to job search behaviour and the motivation, or lack of, when engaging in the behaviour. Gagne et al. (2005) also discuss the importance of supporting autonomy within an individual through those close to the individual. This will be discussed later in the chapter. Gagne et al. (2005) concluded that many studies have confirmed SDT’s concepts of intrinsic and extrinsic motivators as aids in motivation. Additionally, it is apparent that in some circumstances it is satisfactory to support not only autonomous intrinsic behaviour, but autonomous extrinsic behaviour in order to gain motivation (Gagne et al., 2005). However, this is sometimes detrimental to motivation and psychological well-being so must be treated cautiously (Gagne et al., 2005).

====Self determination theory in comparison with expectancy-value theory====
[[File:List-372766 640.jpg|thumbnail|left|''Figure 6.'' Controlled motivation correlates negatively with psychological well-being{{fact}}.]]
An interesting study by Vansteenkiste et al. (2005) compared expectancy-value theory and self-determination theory in order to understand unemployed individual’s experience of job search behaviour and well-being. Feather and Brian (1986, as cited in Vansteenkiste et al., 2005) stated in a previous study that it would be of some value for future studies to explore the domain of work motivation whilst using more than one theory. Hence, Vansteenkiste et al’s (2005) study was conducted. This study used 481 participants and used questionnaires in the form of the Satisfaction with Life Survey (Diener, Emmons, Larsen &amp; Griffen, 1985, as cited in Vansteenkiste et al, 2005), the General Health Questionnaire (Goldberg, 1978, as cited in Vansteenkiste et al., 2005), a measure of unemployed individual’s past job search behaviour as well as an assessment of the individual’s feelings of meaningless and social isolation (Vansteenkiste et al., 2005). In line with EVT, the first hypotheses{{spelling}} tested was that individual’s that{{grammar}} hold high value on gaining employment would negatively correlate with low levels of well-being. In addition, they hypothesised that having both high levels of expectancy and value on gaining employment would correlate positively to job search behaviour, and would also reduce the effects of unemployment on an individual’s well-being. In line with SDT Vansteenkiste et al. (2005) also hypothesised that autonomous motivation would predict positive job searching motivation as well as controlled motivation to negatively predict job search behaviour. In addition, Vansteenkiste et al. (2005) predicted controlled motivation to predict negative overall feelings and well-being during unemployment. They then used the results of these to compare the two theories (SDT &amp; EVT) in relation to job search behaviour (Vansteenkiste et al, 2005).
 
The results of this comparative study proved Vansteenkiste et al’s (2005) hypotheses to be correct. The study found that individual’s with both autonomous motivation and high value on employment have greater job search motivation (Vansteenskiste et al., 2005). Amongst other statements, Vansteenkiste et al. (2005) stated that it was those individuals whom held high expectancy and value on employment who correlated with negative psychological well-being. This statement claims that because the unemployed individual holds high expectations and value on being employed, the individual is more likely to hold feelings of worthlessness and other negative feelings. Overall, this study is valuable in understanding the effects unemployment has on both job search behaviour and psychological well-being and bears the question as to whether using more than one theory for explanations as to the ‘whys’ and ‘why nots’ of job search behaviour in unemployed individuals is effective or not.

==Supporting job search motivation==

In line with both SDT and the research that has been carried out on job search behaviour, we can see that autonomy and intrinsic motivation are an important part of gaining levels of motivation to assist in job search behaviour. The following section will explain what the unemployed individual can do, and what the people surrounding the unemployed can do to assist in achieving high levels of job search behaviour. Ryan and Deci (2000) wrote an article on the facilitation of motivation and well-being. Among many statements within the article, they speak of motivation being able to grow only if the circumstances allow it (Ryan &amp; Deci, 2000). As in the SDT Ryan &amp; Deci (2000) state that there needs to be supporting components of each of the three psychological needs in order to gain optimal motivation. Consequently, studies have shown that support structures such as positive performace feedback (Ryan &amp; Deci, 2000) and opportunities of choice (Stone, Deci &amp; Ryan, 2008) correlate positively with autonomous and intrinsic motivation.

====Supporting autonomy====
Deci (2012) spoke briefly about SDT and went on to speak about the importance of autonomy support in order to provide an environment of which can support an individual's motivation. Click here to watch Ed Deci speak about SDT and autonomy support. Deci (2012) stated several points as to which should be implemented when attempting to support autonomy. These are:
*'''Understanding''' the individual's '''perspective'''
*Provide '''choice''' to the individual
*'''Support''' the individual's '''exploration''' of different routes
*&quot;Provide '''meaningful rationale''' &quot; (Deci, 2012).

To extend on these points, Stone, Deci &amp; Ryan (2008) wrote an article based upon several steps in creating and supporting autonomous motivation using the basis of SDT. Although this particular article has a basis of companies and employees within them, the information can be easily translated into ‘how to create and support autonomy amongst unemployed individuals'. Firstly, they spoke of asking open-ended questions. In the context of unemployment, this would relate to the unemployed individual’s family and friends asking open ended questions such as, ‘tell me what avenues you have used so far in your job search.’ This opens up a conversation and assists in the individuals '''exploration''' of different routes. Whereas if a family member asks, ‘haven’t you looked on Z website?’, the unemployed individual may interpret this as controlling, and therefore is not supportive of their autonomous motivation (Stone et al., 2008). Sequentially to this, Stone et al., (2008) discuss the importance of active listening to autonomy support. Not only does this provide a feeling of value, but also acknowledges your understanding of the individual’s '''perception''' of the situation (Stone et al., 2008, Ryan &amp; Deci, 2000). 

Furthermore, Stone et al, (2008) discuss the importance of offering '''choice'''. In the context of unemployment, it is essential to allow the individual to have a choice while searching for a job. Ie. When they search, how they search etc. To aid in the individuals choice, it can be useful to provide a '''meaningful rationale''' (Stone et al., 2008, Ryan &amp; Deci, 2000).

Another idea Stone et al., (2008) suggest in line with SDT is to minimise extrinsic motivation in order to raise intrinsic motivation is to minimise external rewards such as rewards and pay rises. Studies have found that where extrinsic motivation is in the forefront of an individual’s motivation, intrinsic motivation fades, making the quality of motivation lower (Stone et al,, 2008, Ryan &amp; Deci, 2000). In addition, studies have found that “the more strongly people value money, the poorer their psychological health” (Grouzet et al., 2002; Ryan et al., 1999 as cited in Stone et al., 2008).

==Conclusion==
Numerous studies show the effects of unemployment on psychological health as well as job search behaviour{{fact}}. It is clear that some of the psychological effects of unemployment are depression, anxiety, low levels of self-esteem and in some cases suicide. Along with unemployment, these psychological effects have a major effect on motivation levels and the extent to which the individual engages in job search behaviour. This chapter has discussed the theories of learned helplessness and expectancy-value, however focused on self-determination theory in order to identify and understand the effects of which unemployment has on job search behaviour. After analyses of the literature and research in this field, it is apparent that the effect unemployment has on job search behaviour and consequently overall motivation is severe{{fact}}. As a result of unemployment, well-being of the individual is low, causing job search behaviour to be minimal{{fact}}. This is because of many aspects including lack of autonomy, relatedness and competence. In line with SDT, without these three psychological needs, motivation, in this case job search behaviour becomes low. 

*'''Autonomy''' is lacking whilst unemployed because the individual feels they have no choice but to search for a job, thus engaging in controlled motivation.
*'''Relatedness''' may lack due to unemployment because the individual is no longer a part of a group ie. a workplace.
*'''Competence''' will lack the longer an individual is searching for a job and not gaining employment as this causes the belief of being able to act to gain a job to become non-existent. 

Therefore, when a loved one is unemployed it is apparent that supporting their autonomous motivation is critical, as is avoiding becoming controlling. The effect of unemployment on job search motivation is that unless the three psychological needs, in particular autonomy, are fostered, job search motivation diminishes because of the lack of the three psychological needs. 

==Quiz: test your knowledge==
&lt;quiz display=simple&gt;
{According to SDT, which of the following is '''not''' one of the three psychological needs?
|type=&quot;()&quot;}
+ A. Connectedness
- B. Relatedness
- C. Autonomy
- D. Competence

{Which of the following are psychological consequences of unemployment?
|type=&quot;()&quot;}
- A. Obesity
- B. Autonomy
+ C. Depression and Anxiety
- D. Connectedness

{According to Deci (2012), which of the following is one aspect to supporting autonomy?
|type=&quot;()&quot;}
- A. Offering a cash reward
- B. Controlling the behaviour with statements
+ C. Providing a meaningful rationale
- D. Drinking a coffee

{__________ is an example of an open-ended question.
|type=&quot;()&quot;}
- A. Do you want to get a job?
+ B. What avenues have you taken in your job search?
- C. Have you tried Z website to search for a job?
- D. Have you got a resume?
&lt;/quiz&gt;

==See also==
*[[Motivation_and_emotion/Book/2013/Avoidance_motivation|Avoidance Motivation]] (2013 Book Chapter)
*[[Motivation_and_emotion/Book/2011/Avoidance_motivation|Avoidance Motivation]] (2011 Book Chapter)
*[[Motivation_and_emotion/Book/2011/Self-determination_theory|Self-determination theory]] (2011 Book Chapter)
*[[Motivation_and_emotion/Book/2013/Workplace_motivation|Workplace motivation]] (2013 Book Chapter)
*[[Wikipedia:Self-determination theory|Self-determination theory (Wikipedia)]]

==References==
{{Hanging indent|1=
Australian Bureau of Statistics (2014). National Survey of Mental Health and Wellbeing: Summary of Results, 2007 Retrieved from: http://www.abs.gov.au/ausstats/abs@.nsf/Latestproducts/4326.0Main%20Features32007?open

Australian Psychological Society (2013). ''Stress and wellbeing in Australia survey: 2013'' Retrieved from: http://www.psychology.org.au/Assets/Files/Stress%20and%20wellbeing%20in%20Australia%20survey%202013.pdf

Creed, P. A., &amp; Macintyre, S.R. (n.d.). The relative effects of deprivation of the latent and manifest benefits of employment on the wellbeing of unemployed people. Retrieved from:  http://www98.griffith.edu.au/dspace/bitstream/handle/10072/3989/15601.pdf?sequence=1

Deci, E. L., &amp; Ryan, M. R. (2000). Intrinsic and Extrinsic Motivations: Classic Definitions and New Directions. ''Contemporary Educational Psychology, 25'', 54-67. doi: 10.1006/ceps.1999.1020 

Deci, E.L. (2012, August 14). ''Promoting motivation, health, and excellence: Ed Decit at TEDxFlourCity'' Retrieved from: http://tedxtalks.ted.com/video/Promoting-Motivation-Health-and;search%3Apromoting%20motivation

Gagne, M., &amp; Deci, L. D. (2005). Self-determination theory and work motivation. ''Journal of Organizational Behavior, 26'', 331-362. doi: 10.1002/job.322

Klehe, U.C., &amp; van Hoofte, E. A. J. (2001). Job search behavior as a multidimensional construct: A review of different job search behaviors and sources. '' Oxford Handbook of Job Loss and Job Search'' (in press). New York: Oxford University Press. Retrieved from: file:///C:/Users/Gareth/Downloads/Chapter%20Different%20Job%20Search%20Behaviors%20in%20press.pdf

Kuhnert, K. W. (1989). The latent and manifest consequences of work. ''Journal of Psychology, 123'', 417-428. Retrieved from: http://zh9bf5sp6t.search.serialssolutions.com/?ctx_ver=Z39.88-2004&amp;ctx_enc=info%3Aofi%2Fenc%3AUTF-8&amp;rfr_id=info:sid/summon.serialssolutions.com&amp;rft_val_fmt=info:ofi/fmt:kev:mtx:journal&amp;rft.genre=journal&amp;rft.pub=Plenum&amp;rft.issn=0146-7239&amp;rft.eissn=1573-6644&amp;rft.externalDBID=n%2Fa&amp;rft.externalDocID=3496387&amp;paramdict=en-US

Murphy, G.C., &amp; Athanasou, J. A. (1999). The effect of unemployment on mental health. ''Journal of Occupational &amp; Organizational Psychology, 72'', 83-99. doi: 10.1348/096317999166518

Pinder, C. C. (1984). Work Motivation: theory, issues, and applications. Glenview, United States of America: Scott, Foresman and Company.

Rodriguez, Y. M. (1997). Learned helplessness or expectancy-value? A psychological model for describing the experiences of different categories of unemployed people. ''Journal of Adolescence, 20'', 321-332. doi: 10.1006/jado.1997.0088

Ryan, M. R. &amp; Deci, E. L.(2000) Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being. ''American Psychologist, 55'', 68-78. Retrieved from: http://mofetinternational.macam.ac.il/jtec/Documents/Self-Determination%20Theory%20and%20the%20Facilitation%20of%20Intrinsic%20Motivation,%20Social%20Development,%20and%20Well-Being.pdf

Stone, D.N., Deci, E.L., &amp; Ryan, R.M. (2009). Beyond talk: creating autonomous motivation through self-determination theory. ''Journal of General Management, 34 (3)'' p75-91. Retrieved from: http://zh9bf5sp6t.search.serialssolutions.com/?ctx_ver=Z39.88-2004&amp;ctx_enc=info%3Aofi%2Fenc%3AUTF-8&amp;rfr_id=info:sid/summon.serialssolutions.com&amp;rft_val_fmt=info:ofi/fmt:kev:mtx:journal&amp;rft.genre=article&amp;rft.atitle=Beyond+talk%3A+creating+autonomous+motivation+through+self-determination+theory&amp;rft.jtitle=Journal+of+General+Management&amp;rft.au=Stone%2C+Dan+N&amp;rft.au=others&amp;rft.date=2009&amp;rft.issn=0306-3070&amp;rft.volume=34&amp;rft.issue=3&amp;rft.spage=75&amp;rft.externalDBID=SCANFILE&amp;rft.externalDocID=Beyond_talk_creatin20090154&amp;paramdict=en-US

Vansteenkiste, M., Lens. W., De Witte, H., &amp; Feather, N. T. (2005) Understanding unemployed people's job search behaviour, unemployment experience and well-being: A comparison of expectancy-value theory and self-determination theory. ''British Journal of Social Psychology, 25'', 269-287. doi: 10.1348/014466604X17641.

Vansteenkiste, M., Lens, W., De Witte, S., De Witte, H., &amp; Deci, E, L. (2004). The ‘why’ and ‘why not’ of job search behaviour: Their relation to searching, unemployment experience, and well-being. ''European Journal of Social Psychology, 34'', 345-363. doi: 10.1002/ejsp.202.
}}

==External links==
*[http://tedxtalks.ted.com/video/Promoting-Motivation-Health-and;search%3Apromoting%20motivation Tedx Talk - Deci speaks about supporting autonomy]

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__TOC__

==Overview==
[[File:Dr Seuss - Brevard Zoo, Viera FL - Flickr - Rusty Clark.jpg|thumb|200px|Figure 1. “Unless someone like you cares a whole awful lot, Nothing is going to get better. It's not.” ― Dr. Seuss, The Lorax]]

Why do some people use their energy, time and other resources to actively fight for and raise awareness about environmental, human and other animal issues? Whilst others don't seem to care at all?

This chapter will explore some of the motivations that people have to become activists in different social justice movements. The following information can be used in two different ways. Firstly, it can be used by individuals to learn about the different motivating factors and theories, which in turn will hopefully motivate their own activism. And secondly, it can be used by organisers and recruiters of social justice groups to help shape their recruitment processes and education campaigns to get more people involved.

== The who, what and why of activism ==
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=== What is activism? ===
&lt;div style=&quot;padding:0.4em 1em 0.3em 1em;&quot;&gt;

;&lt;blockquote&gt;“Activist orientation is defined as an individual’s developed, relatively stable, yet changeable orientation to engage in various collective, social-political, problem-solving behaviours spanning a range from low-risk, passive, and institutionalized acts to high-risk, active, and unconventional behaviours.” Corning &amp; Myers (2002)&lt;/blockquote&gt;

The comprehensive nature of Corning and Myers’s definition demonstrates that activism is not just one thing. Our society is based on large scale oppression (Maxey, 1999) which favours the powerful few over the vulnerable many. The many include not only humans but other animals and the environment too. On our planet [http://www.adaptt.org/killcounter.html billions of animals are killed daily] for consumption by a small portion of the population while over 800 million people live on less than $1.25 a day and have difficulty accessing food and clean drinking water (United Nations Development Project, 2015). Meanwhile, reports from the Intergovernmental Panel on Climate Change (2014) have announced that all food security across the planet is at risk due to climate change.

Maxey (1999) suggests that the overwhelming level of oppression can be disempowering and yet there are still activists who take a stand every day. Some activists aim to reform current governmental policies in hopes of improving the welfare of people or other animals in the immediate future while other more transformative activists (Zoller, 2005) aim to dismantle social structures in hopes of challenging oppression at its cause.

Activism encompasses a wide range of behaviours (Faver, 2001) which includes everything from writing and calling politicians about local, national or international issues, taking legal action, organising or attending rallies, demonstrations and protests, writing submissions, educating and awareness raising, signing a petition, boycotting an event or product to wearing a t-shirt with a social justice slogan on it.

[[Image:VeganRallyEvolve.jpg|thumb|220px|Figure 4. Animal Rights activism.]]
[[Image:Acto cerremos guantanamo.jpg|thumb|220px|Figure 5. Human Rights activism.]]

&lt;/div&gt;How to tell if you're an activist?&lt;quiz display=&quot;simple&quot;&gt;
{Have you ever...
|type=&quot;[]&quot;}
+ Written to your local MP about an issues that you were concerned about?
+ Attended a rally or a protest?
+ Campaigned for a political party?
+ Written to local media about an issues that concerned you?
+ Volunteered for a social justice organisation (environmental, animal or human)?
+ Helped organise an event which raised community awareness about a social issue?
+ Chosen to live your life in a way that reduces your carbon footprint (i.e. you eat a plant-based diet, ride a bike or walk instead of drive etc)?
+ Self-identified as an activist?
&lt;/quiz&gt; The more questions you answered 'yes' to the more likely you're an activist!

| style=&quot;padding:0em 0.5em 0em 0.5em;&quot; |

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=== Who are activists? ===
[[Image:Activist in Turkey believes in Health as a Human Right. (15438670950).jpg|thumb|200px|Figure 2. Activists work towards social change.]]

Being an activist can range from being a paid professional, such as working for Amnesty International (Rodgers, 2010), working as an alternative journalist (Harcup, 2011), through to volunteering for a local community organisation. You can even be an activist by yourself! The main element that constitutes activism is taking action to ''affect social change'' (Faver, 2001; Gilster, 2012; Lindblom &amp; Jacobsson, 2014; Zoller, 2005). This can be in relation to the environment, humans, other animals or in many cases, a combination of all three.

Activists may break social norms in order to behave in ways which better align with their own moral code and while this can reduce an individual’s cognitive dissonance it can also lead to alienation from society (Lindblom &amp; Jacobsson, 2014). For example, a person who believes that non-human animals do not exist for human pleasure may not eat, use or wear animals and animal by-products. This form of lifestyle activism (Cherry, 2015), known as veganism, is a challenge to mainstream beliefs and societal norms in regards to non-human animals.
|-

| style=&quot;padding:0em 0.5em 0em 0.5em;&quot; |

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=== Examples of activism ===

# With more data being collected about the direct impacts of climate change on human food production and weather patterns (Climate Institute, 2015) more people are starting to care about environmental issues (Mihaylov &amp; Perkins, 2015). According to Measham &amp; Barnett (2008) environmental activism involves activities which range from educating the public about what is happening to the environment and its impact on humans and other animals, especially wildlife, to full-scale environmental protection which involves stopping the destruction of forests, waterways and wildlife habitats. People from all walks of life get involved with environmental activism as demonstrated by the [http://www.lockthegate.org.au/ Lock the Gate Campaign]. The campaign aims to stop unsafe mining in Australia with an emphasis on blocking fracking for coal seam gas and has attracted the support of thousands of ordinary people as well as influential Australians like Alan Jones (ABC, 2014).
# Animal rights activism has a wide range of activities that activists participate in to try and reduce or stop the suffering of non-human animals. These activities can range from talking to someone about what happens to animals in scientific laboratories or in slaughterhouses to taking direct action against businesses who profit from animal cruelty via protests, demonstrations and economic sabotage. Again, while there are stereotypes of what an animal rights activists looks and acts like in reality there are people from all walks of life who get involved in the movement.
# Human rights activists range from people who work in structured international organisations like Amnesty to people who are involved with local charities like the Salvation Army who help feed and clothe the homeless. Other forms of human rights activism which people may not recognise as activism is health activism (Zoller, 2005) which involves raising awareness about health issues in the community especially in regards to minorities such as women, Indigenous and LGBITQ peoples as well as health care reform and illness advocacy. In Australia, a current pressing human rights issue is that of asylum seekers coming to Australia via boats and being held in mandatory detention centres offshore. Many groups have banded together to hold national rallies (ABC, 2015) to let the government know that most Australians do not support detention centres for refugees further demonstrating that anyone can be a human rights activist.
|}

==Motivational theories==
There are many motivating factors when looking at why people get involved with activism. The following section on motivational theories reflects this:

Note: Please note this is not an exhaustive list of theories.

=== Self-Determination Theory ===
[[Image:SelfDeterminationTheory.png|thumb|360px|Figure 7. Self Determination Theory.]]

Self-Determination Theory (SDT) is the study of &quot;people's inherent growth tendencies and innate psychological needs that are the basis for their self-motivation and personality integration, as well as for the conditions that foster those positive processes&quot; (Ryan &amp; Deci, 2000, p. 68). Ryan &amp; Deci (2000) believe that intrinsic motivation is the key to people's ability to seek challenges, grown and learn in the right social-environmental conditions. More specifically, Cognitive Evaluation Theory (CET) a subset of SDT, found that intrinsic motivation was determined by social environments which either supported or inhibited three psychological needs; autonomy, competence and relatedness. A second subset theory of SDT called Organismic Intergration Theory (OIT) relates to the intergration{{spelling}} and internalisation of values which lead to external motivation to achieve autonomy, competence and relatedness. In their meta-analysis Ryan &amp; Deci (2000) argue that the pursuit of intrinisic{{spelling}} aspirations which lead to autonomy, competence and relatedness can positively affect mental health and well-being with the inverse occurring i.e. psychopathology and ill-being if these psychological needs are not met.

In regards to activism, if a person holds social justice for animals, the environment and/or humans as an internal aspiration because it fulfils{{spelling}} their psychological needs of relatedness then SDT would say that they are intriniscally{{spelling}} motivated toward that goal and are compelled to achieve it for their own mental health. Further, OIT demonstrates how people can encourage others to internalise values regarding the environment, humans and other animals which fosters external motivations toward activism too.

=== Moral schemas ===
Farrell’s (2011) research on environmental activist’s{{grammar}} motivations and moral schemas led him to categorise his study's sample into different groups; the unenchanted and the enchanted (which he then broke down further into the creational and the intrinsic). These three categories represent the different moral schemas people held of the environment. The enchanted groups both held the internalised schemas of the environment which gave it significance in their world view. The difference between the two groups was that the enchanted intrinsics held the view that nature was in and of itself sacred while the enchanted creationalists believed that the environment was sacred due to spiritual/religious reasons. While the unenchanted believed the environment was important but didn’t hold a sacred view of it{{grammar}}.

The results of the study showed that the enchanted intrinsics, those who believe the environment is sacred in itself, were more likely to be involved in environmental activism (for example, they belonged to an environmental group or donated money to an environmental cause). The research suggests that people who have an internalised moral schema which differentiates the environment as something sacred may feel a greater moral obligation toward it. Berenguer's (2010) research also found that people who held higher empathic values for animals and humans were more likely to be pro-environment. People who hold these enchanted moral schemas and empathic values, therefore, may be more intrinsically motivated to act in ways which support/protect the environment and therefore more likely to become an activist.

=== Theory of planned behaviour &amp; value-belief-norm ===

[[Image:Theory of planned behavior.png|thumb|360px|Figure 8. Theory of planned behaviour.]]

Ajzen's (1991) theory of planned behaviour (TPB) suggests that the combination of:
# attitudes a person holds toward a behaviour,
# perceptions a person has about the norms regarding the behaviour, as well as
# the perceived control a person has over the behaviour
are the best predictors as to whether a person will carry out said behaviour. Further, Stern et al.'s (1999) value-belief-norm theory (VBN) has demonstrated that there is a causal process where personal beliefs about the environment precede behavioral norms which precede pro-environmental behaviors. In their own research, Oreg &amp; Katz-Gerro (2006) combined the TPB, VBN and two nation-level values (harmony and post-materialism) to create a test model which successfully predicted pro-environmental behaviours cross-culturally. The specific results from their study demonstrated that the psychological constructs which signify the way people think and feel toward the environment motivates their behaviour.

=== Ethic of care ===
Faver’s (2001) qualitative research explored the motivations behind women’s social activism, morals and spirituality and found three overarching motivational themes:
# to ensure rights
# to fulfill responsibilities, and
# to restore relationships and build community

These three themes are the basis of a theory known as the Ethic of Care. Although a criticism of Faver’s study is that her sample of 50 women was quite homogeneous which limits its generalisability of this study{{grammar}}. Other research, however, has also found that feelings of [[Motivation and emotion/Book/2014/Altruism and empathy|empathy]] and compassion which translate into a concern for others is a major motivating factor in carrying out specific forms of humanitarian activism (Omoto, Snyder &amp; Hackett, 2010).

The theory of ethic of care was originally proposed by Carol Gilligan and has now been built upon by many feminist scholars to not only encompass compassion for humans but also for the environmental and non-human animals (Donovan &amp; Adams, 2007). The theory is “a more flexible, situational and particularised ethic” (Donovan &amp; Adams, 2007) which suggests that we have a moral responsibility toward others which is more about how we relate to others and the contexts in which those relations occur. In other words, an ethic of care sees individuals as being important in and of themselves and situation dependent which means there are no hard and fast rules to regulate moral considerations. This is contrast to many rights-based theories which have rules and regulate who has access to rights dependent on certain features rather than situations (Donovan &amp; Adams, 2007).

Specifically in animal rights, the feminist ethic of care theory developed as a response to the anti-emotional and pro-rational approaches favoured by many of the movements{{grammar}} male (alleged) founders and leaders such as Peter Singer and Tom Regan (Donovan, 2007). Their rational theories as to why we should not use animals (utilitarianism for Singer and moral rights for Regan) were seen as logical and understandable but as Jasper (1998) puts it cognitive agreement alone does not necessarily translate into action and people still continue to use and abuse animals.

It is because of this disregard of spiritual, emotional and relational connections between humans that feminist animal rights activists argue against the use of rational arguments to motivate people to care or act for animals (or the environment or humans). The results from Faver’s study demonstrated that an ethic of care theory creates a sense of interconnectedness and a moral obligation to others which acts as a motivational factor in leading people to activism and sustaining their role within social movements.

=== Emotions ===
Jasper (1998) posits that it is emotion that different social movement group organisers appeal to when trying to motivate potential volunteers, members, voters or activists to action. Different emotions motivate people into different forms of action. For example, Rodgers (2010) in her study of paid employees of Amnesty International found that the many workers were motivated to do their work by powerful emotions such as guilt. Their feelings of guilt arose when they compared their lives to that of their clients whom they researched or advocated on behalf of and often led to feelings of distress. Further demonstrating that emotions play a role in motivation across a range of activists fields, Askins (2009) an academic-activist states that it is the intense range of emotions she feels about social and environmental issues which fuels her passion to make social change via her field.

Jasper (1998) further suggests that shock tactics can be used to cause outrage and alarm among people in order to educate and motivate them to action. A study by Miller and Krosnick (2004) found, however, that by giving people opportunities to help rather than causing fear-induced responses campaigns were more likely to engage people and motivate them to take social action.

An example of this occurred in 2011 when the Australian news program ''Four Corners'' ran an exposé on the Australian live cattle trade using footage taken by activists from Animals Australia's investigation team showing animals being treated horrifically by slaughterhouse workers (Animals Australia, 2015). The program sparked a national campaign to have the Australian live export industry shut down and inspired thousands of ordinary Australian people to take the streets in protest (ABC, 2011). While the shocking nature of the footage filled many people with sadness and despair it also filled many people with outrage and anger. Animals Australia and supporting animal rights organisations utilised these emotions to encourage people to express these emotions via rallies held in many cities and towns across the country in the weeks following the news program as well as toward their local politicians in the form of online petitions and form letters. These opportunities for action, immediately following the intense affect caused by the discovery of animal mistreatment, led to an unprecedented number of people to stand up for animals in Australia and become activists, even if just temporarily.

Finally Jasper (1998) suggests that is due to emotions that joining social movement groups or becoming part of a network of like-minded individuals to fight for a common good can be pleasurable in and of itself and it is this pleasure which motivates people to get involved. For example, people who believe that animals should not be eaten by humans join vegan social groups or societies to enjoy cruelty-free eating whilst socialising and feminist groups often create safe spaces for women only to socialise to challenge male-dominated social spaces.

=== Barriers to activism ===
Becoming involved in activism is not necessarily as simple as finding what motivates someone and then encouraging that factor. Unfortunately for many motivated people there are barriers in the way to getting involved with activism. As Klandermans &amp; Oegema (1987) state motivation and barriers interact to activate participation. Barriers may include, but are not limited to;
* physical ability - e.g. some people are not able to part in rallies or fundraising events which involve walking or running due to physical disabilities
* socio-economic status - e.g. joining social events or being a part of actions can cost money and/or time which some people do not have (Miller &amp; Krosnick, 2004)
* lack of social networks - e.g. people may be motivated to get involved in social actions like protests but may lack the social networks to feel comfortable attending (Klandermans &amp; Oegema, 1987)
* pessimism - e.g. some people who are sympathetic to social causes are unwilling to actively join a movement because they are unable to see how individuals can make a difference (Swank &amp; Fahs, 2011)

==Conclusion==
Activism is good for one's well-being (Klar &amp; Kasser, 2009) and empowering (Gilster, 2012) while also working toward a common good for the environment, humans and other animals. While there are many motivating factors to becoming involved with activism there are also some barriers which may limit participation, however, there are ways to overcome them.

[[File:Animal Rights Activists Protest at Circus.jpg|thumb|350px|Figure 9. Activists participating in a protest against the use of animals in entertainment.]]

=== To become an activist ===
* Work out what you care about how you want to make a difference – environment, animals, humans, or even better, all three!?
* If you'd like to join a group then search online, especially in local media, for different organisations or groups that are in your local area. If you can't find any around or any that interest you then check for online groups - Facebook is a great place to find grass-roots community organisations and social groups.
* Work out what your assets are:
** Are you keen and full of energy to get involved hands-on or would rather something low-key and behind the scenes (or both!)?
** Do you have spare time to offer help or are you strapped for time but could help out in financial or networking ways?
** Do you have skills you could offer like IT, artistic or planning skills?
*** These are just a few of the things you could offer but most organisations or groups are just keen on having new people with energy and positivity get involved!
* Contact the group and see if there are specific ways you can get involved or turn up to their next event or meeting.
* If you can't find a group you want to join or there isn't a group doing what you want to do then start your own!

=== To encourage activists ===
* Collom (2011) suggests that tailoring the activities or tasks that you need done to the actual people who you are targeting will lead to more satisfied volunteers and a higher chance of volunteers becoming more involved in the future.
* Jasper (1998) believes to successfully recruit people it is very important how an organisation frames and convinces its potential recruits of its goals, the tools in which it will achieve its goal/s and why they recruits should get involved. His research shows that an organisations{{grammar}} claims should be empirically sound as well as complementary to the people they are trying to recruit. i.e asking an impoverished community to fund an expensive campaign to fix up the local tennis courts would not be a well-suited or successful recruitment process.
* Oreg &amp; Katz-Gerro (2006) argue that instead of increasing people's knowledge about environmental (or animal or human) issues as most educational programs aim to do activists need to focus their educational programs on changing cultural values in order to get people to change their behaviours regarding the environment, animals and humans and thus become more involved in social justice movements.

==See also==
[[w:Environmentalism|Environmentalism]]

[[w:Animal_rights|Animal Rights]]

[[w:Human_rights|Human Rights]]

==References==
{{Hanging indent|1=
ABC. (2011, August 14). Thousands march against live animal exports. ''ABC''. Retrieved from http://www.abc.net.au/news/2011-08-14/thousands-march-to-protest-live-export/2838772

Animals Australia. (2011. May 30). Indonesian live export investigation on Four Corners damning. ''Animals Australia''. Retrieved from http://www.banliveexport.com/features/live-export-investigation-on-four-corners.php

Ajzen, I. (1991). The theory of planned behavior. ''Organizational Behavior and Human Decision Processes, 50,'' 179-211.

Askins, K. (2009). ‘That's just what I do’: Placing emotion in academic activism. ''Emotion, Space and Society'', ''2'', 4-13. http://dx.doi.org/10.1016/j.emospa.2009.03.005

Berenguer, J. (2008). The Effect of Empathy in Environmental Moral Reasoning. ''Environment and Behavior'', ''42'', 110-134.  http://dx.doi.org/10.1177/0013916508325892

Cherry, E. (2014). I Was a Teenage Vegan: Motivation and Maintenance of Lifestyle Movements. ''Sociological Inquiry'', ''85'', 55-74. http://dx.doi.org/10.1111/soin.12061

Collom, E. (2011). Motivations and Differential Participation in a Community Currency System: The Dynamics Within a Local Social Movement Organization. ''Sociological Forum'', ''26'', 144-168. http://dx.doi.org/10.1111/j.1573-7861.2010.01228.x

Corning, A., &amp; Myers, D. (2002). Individual Orientation Toward Engagement in Social Action. ''Political Psychology'', ''23'', 703-729. http://dx.doi.org/10.1111/0162-895x.00304

Donovan, J. (2007) Animal Rights and Feminist Theory. In J. Donovan &amp; C. Adams (Eds). ''The Feminist Care Tradition in Animal Ethics (pp. 58-86).'' New York: Columbia University Press.

Donovan, J., &amp; C. Adams. (2007). ''The Feminist Care Tradition in Animal Ethics.'' New York: Columbia University Press.

Farrell, J. (2011). Environmental Activism and Moral Schemas: Cultural Components of Differential Participation. ''Environment and Behavior'', ''45'', 399-423. http://dx.doi.org/10.1177/0013916511422445

Faver, C. (2001). Rights, Responsibility, and Relationship: Motivations for Women's Social Activism. ''AFFILIA'', ''16'', 314-336. http://dx.doi.org/10.1177/088610990101600304

Gilster, M. (2012). Comparing Neighborhood-Focused Activism and Volunteerism: Psychological Well-Being and Social Connectedness. ''Journal of Community Psychology'', ''40'', 769-784. http://dx.doi.org/10.1002/jcop.20528

Harcup, T. (2011). Alternative Journalism as Active Citizenship. ''Journalism'', ''12'', 15-31. http://dx.doi.org/10.1177/1464884910385191

Jasper, J. (1998). The Emotions of Protest: Affective and Reactive Emotions in and around Social Movements. ''Sociological Forum'', ''13'', 397-424.

Klandermans, B., &amp; Oegema, D. (1987). Potentials, Networks, Motivations, and Barriers: Steps Towards Participation in Social Movements. ''American Sociological Review'', ''52'', 519. http://dx.doi.org/10.2307/2095297

Klar, M., &amp; Kasser, T. (2009). Some Benefits of Being an Activist: Measuring Activism and Its Role in Psychological Well-Being. ''Political Psychology'', ''30'', 755-777. http://dx.doi.org/10.1111/j.1467-9221.2009.00724.x

Lindblom, J., &amp; Jacobsson, K. (2013). A Deviance Perspective on Social Movements: The Case of Animal Rights Activism. ''Deviant Behavior'', ''35'', 133-151. http://dx.doi.org/10.1080/01639625.2013.834751

Matsuba, M., &amp; Pratt, M. (2013). The Making of an Environmental Activist: A Developmental Psychological Perspective. ''New Directions for Child and Adolescent Development'', ''2013'', 59-74. http://dx.doi.org/10.1002/cad.20049

Maxey, I. (1999). Beyond boundaries? Activism, Academia, Reflexivity and Research. ''Area'', ''31'', 199-208. http://dx.doi.org/10.1111/j.1475-4762.1999.tb00084.x

Measham, T., &amp; Barnett, G. (2008). Environmental Volunteering: Motivations, Modes and Outcomes. ''Australian Geographer'', ''39'', 537-552. http://dx.doi.org/10.1080/00049180802419237

Mihaylov, N., &amp; Perkins, D. (2015). Local Environmental Grassroots Activism: Contributions from Environmental Psychology, Sociology and Politics. ''Behavioral Sciences'', ''5'', 121-153. http://dx.doi.org/10.3390/bs5010121

Miller, J., &amp; Krosnick, J. (2004). Threat as a Motivator of Political Activism: A Field Experiment. ''Political Psychology'', ''25'', 507-523. http://dx.doi.org/10.1111/j.1467-9221.2004.00384.x

Omoto, A., Snyder, M., &amp; Hackett, J. (2010). Personality and Motivational Antecedents of Activism and Civic Engagement. ''Journal of Personality'', ''78'', 1703-1734. http://dx.doi.org/10.1111/j.1467-6494.2010.00667.x

Oreg, S., &amp; Katz-Gerro, T. (2006). Predicting Proenvironmental Behavior Cross-Nationally: Values, the Theory of Planned Behavior, and Value-Belief-Norm Theory. ''Environment and Behavior, 38,'' 462-483. http://dx.doi.org/10.1177/0013916505286012

Reinfrank, A. (2015, October 11). Rally opposing detention of asylum seekers draws huge crowds in Canberra's CBD. ''ABC.'' Retrieved from http://www.abc.net.au/news/2015-10-11/hundreds-rally-in-support-of-asylum-seekers-in-canberra/6845020

Rodgers, K. (2010). ‘Anger is Why We're All Here’: Mobilizing and Managing Emotions in a Professional Activist Organization. ''Social Movement Studies'', ''9'', 273-291. http://dx.doi.org/10.1080/14742837.2010.493660

Ryan, R., &amp; Deci, E. (2000). Self-Determination Theory and the Facilitation of Intrinsic Motivation, Social Development, and Well-Being. ''American Psychologist'', ''55'', 68-78. http://dx.doi.org/10.1037//0003-066x.55.1.68

Stern, P. C., Dietz, T., Abel, T., Guagnano, G. A.,&amp; Kalof, L. (1999). A value-belief-norm theory of support for social movements: The case of environmentalism. ''Human Ecology Review, 6,'' 81-97.

Swank, E., &amp; Fahs, B. (2011). Students for Peace: Contextual and Framing Motivations of Antiwar Activism. ''Journal of Sociology &amp; Social Welfare'', ''38'', 111-136.

Taylor, K. (2014, August 4). Drew Hutton and Alan Jones renew old ties for Lock The Gate movement against coal seam gas wells. ''ABC''. Retrieved from http://www.abc.net.au/news/2014-08-04/alan-jones-and-greens-co-founder-reunite-for-csg-fight/5644506

The Climate Institute. (2015, August 10). Media Release - Post-2020 pollution reduction targets announcement a critical opportunity for Abbott government to reflect public sentiment on climate, renewables and carbon pollution. ''The Climate Institute''. Retrieved from http://www.climateinstitute.org.au/articles/media-releases/post-2020-pollution-reduction-targets-announcement-a-critical-opportunity-for-abbott-government-to-reflect-public-sentiment-on-climate,-renewables-and-carbon-pollution.html

United Nations Development Project. (2015). Goal 1: No poverty. Retrieved from United Nations Development Project website: http://www.undp.org/content/undp/en/home/mdgoverview/post-2015-development-agenda/goal-1.html

Veldman, R. (2012). Narrating the Environmental Apocalypse: How Imagining the End Facilitates Moral Reasoning Among Environmental Activists. ''Ethics and the Environment'', ''17'', 1-23. http://dx.doi.org/10.2979/ethicsenviro.17.1.1

Zoller, H. (2005). Health Activism: Communication Theory and Action for Social Change. ''Communication Theory'', ''15'', 341-364. http://dx.doi.org/10.1093/ct/15.4.341
}}

==External Links==
Links to activism resources:

[http://www.amnesty.org.au/actsnsw Amnesty International ACT &amp; NSW]

[http://www.al-act.org Animal Liberation ACT]

[http://www.aycc.org.au/ Australian Youth Climate Coalition (AYCC)]

[http://refugeeaction.org/ Refugee Action Committee Canberra]

[http://www.see-change.org.au/ SEE-Change]

[http://vegact.org.au/ Vegan ACT]

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Activism]]
[[Category:Motivation and emotion/Book/Social psychology]]</text>
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  </page>
  <page>
    <title>Haskell programming in plain view</title>
    <ns>0</ns>
    <id>203942</id>
    <revision>
      <id>2815080</id>
      <parentid>2812678</parentid>
      <timestamp>2026-06-10T19:03:38Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Lambda Calculus */</comment>
      <origin>2815080</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6273" sha1="562nt8nc7q9hm90tsq99nv1dcfk0i3v" xml:space="preserve">
==Introduction==
* Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]])
* Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]])
* Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]])
* Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]])
* Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]])
&lt;/br&gt;

==Applications==
* Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]])
* Bird's Implementation 
:- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]])
:- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]])
:- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]])
:- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]])

&lt;/br&gt;

==Using GHCi==
* Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]])
&lt;/br&gt;

==Using Libraries==
* Library ([[Media:Library.1.A.20170605.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;


==Types==
* Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]])
* TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]])
* Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]])
* Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]])
* Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]])

==Functions==
* Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]])
* Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]])
* Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]])

==Expressions==
* Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]])
* Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]])
* Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;

==Lambda Calculus==
* Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]])
* Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]])
* Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]])
* Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]])
* Encoding Datatypes 
:- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]])
:- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]])
:- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]])
:- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]])
:- Recursions ([[Media:LCal.9A.Recursion.20260608.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]])
&lt;/br&gt;
&lt;/br&gt;

==Function Oriented Typeclasses== 
=== Functors ===
* Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]])
* Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]])
* Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]])

=== Applicatives ===
* Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]])
* Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]])
* Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]])
* Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]])

=== Monads I : Background ===
* Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]])
* Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]])
* Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]])
* Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]])
* IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]])
* Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]])

=== Monads II : State Transformer Monads ===
* State Transformer 
: - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]])
: - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]])
: - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]])

* State Monad 
: - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]])
: - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) 
: - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]])

=== Monads III : Mutable State Monads ===
* Mutability Background 
: - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]])
: - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]])
: - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]])
: - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]])
: - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]])

* Mutable Objects
: - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]])
: - Mutable Data Structures  ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]])

* IO Monad 
: - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]])
: - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]])  
: - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]])  

* ST Monad 
: - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]])
: - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]])
: - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]])

=== Monads IV : Reader and Writer Monads ===
* Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]])
* Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]])
* MonadState Class
:: - State &amp; StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]])
:: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]])
* MonadReader Class
:: - Reader &amp; ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]])
:: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]])
* Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]])

=== Monoid ===
* Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]])

=== Arrow ===
* Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]])
&lt;/br&gt;

==Polymorphism==
* Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]])
&lt;/br&gt;

==Concurrent Haskell ==
&lt;/br&gt;


go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://learnyouahaskell.com/introduction Learn you Haskell]
* [http://book.realworldhaskell.org/read/ Real World Haskell]
* [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material]


[[Category:Haskell|programming in plain view]]</text>
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    <revision>
      <id>2815083</id>
      <parentid>2815080</parentid>
      <timestamp>2026-06-10T19:05:14Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
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      <comment>/* Lambda Calculus */</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6273" sha1="gn8jugv4jxf3ykre8het69681gd65t1" xml:space="preserve">
==Introduction==
* Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]])
* Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]])
* Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]])
* Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]])
* Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]])
&lt;/br&gt;

==Applications==
* Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]])
* Bird's Implementation 
:- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]])
:- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]])
:- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]])
:- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]])

&lt;/br&gt;

==Using GHCi==
* Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]])
&lt;/br&gt;

==Using Libraries==
* Library ([[Media:Library.1.A.20170605.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;


==Types==
* Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]])
* TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]])
* Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]])
* Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]])
* Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]])

==Functions==
* Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]])
* Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]])
* Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]])

==Expressions==
* Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]])
* Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]])
* Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;

==Lambda Calculus==
* Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]])
* Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]])
* Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]])
* Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]])
* Encoding Datatypes 
:- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]])
:- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]])
:- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]])
:- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]])
:- Recursions ([[Media:LCal.9A.Recursion.20260609.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]])
&lt;/br&gt;
&lt;/br&gt;

==Function Oriented Typeclasses== 
=== Functors ===
* Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]])
* Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]])
* Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]])

=== Applicatives ===
* Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]])
* Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]])
* Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]])
* Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]])

=== Monads I : Background ===
* Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]])
* Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]])
* Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]])
* Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]])
* IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]])
* Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]])

=== Monads II : State Transformer Monads ===
* State Transformer 
: - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]])
: - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]])
: - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]])

* State Monad 
: - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]])
: - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) 
: - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]])

=== Monads III : Mutable State Monads ===
* Mutability Background 
: - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]])
: - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]])
: - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]])
: - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]])
: - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]])

* Mutable Objects
: - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]])
: - Mutable Data Structures  ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]])

* IO Monad 
: - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]])
: - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]])  
: - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]])  

* ST Monad 
: - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]])
: - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]])
: - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]])

=== Monads IV : Reader and Writer Monads ===
* Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]])
* Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]])
* MonadState Class
:: - State &amp; StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]])
:: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]])
* MonadReader Class
:: - Reader &amp; ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]])
:: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]])
* Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]])

=== Monoid ===
* Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]])

=== Arrow ===
* Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]])
&lt;/br&gt;

==Polymorphism==
* Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]])
&lt;/br&gt;

==Concurrent Haskell ==
&lt;/br&gt;


go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://learnyouahaskell.com/introduction Learn you Haskell]
* [http://book.realworldhaskell.org/read/ Real World Haskell]
* [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material]


[[Category:Haskell|programming in plain view]]</text>
      <sha1>gn8jugv4jxf3ykre8het69681gd65t1</sha1>
    </revision>
    <revision>
      <id>2815085</id>
      <parentid>2815083</parentid>
      <timestamp>2026-06-10T19:06:18Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Lambda Calculus */</comment>
      <origin>2815085</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6273" sha1="ms7ks3b5w7e4t7zpjbbwnx8166cyvfo" xml:space="preserve">
==Introduction==
* Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]])
* Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]])
* Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]])
* Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]])
* Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]])
&lt;/br&gt;

==Applications==
* Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]])
* Bird's Implementation 
:- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]])
:- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]])
:- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]])
:- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]])

&lt;/br&gt;

==Using GHCi==
* Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]])
&lt;/br&gt;

==Using Libraries==
* Library ([[Media:Library.1.A.20170605.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;


==Types==
* Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]])
* TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]])
* Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]])
* Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]])
* Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]])

==Functions==
* Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]])
* Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]])
* Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]])

==Expressions==
* Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]])
* Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]])
* Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;

==Lambda Calculus==
* Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]])
* Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]])
* Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]])
* Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]])
* Encoding Datatypes 
:- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]])
:- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]])
:- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]])
:- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]])
:- Recursions ([[Media:LCal.9A.Recursion.20260610.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]])
&lt;/br&gt;
&lt;/br&gt;

==Function Oriented Typeclasses== 
=== Functors ===
* Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]])
* Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]])
* Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]])

=== Applicatives ===
* Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]])
* Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]])
* Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]])
* Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]])

=== Monads I : Background ===
* Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]])
* Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]])
* Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]])
* Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]])
* IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]])
* Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]])

=== Monads II : State Transformer Monads ===
* State Transformer 
: - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]])
: - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]])
: - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]])

* State Monad 
: - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]])
: - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) 
: - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]])

=== Monads III : Mutable State Monads ===
* Mutability Background 
: - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]])
: - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]])
: - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]])
: - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]])
: - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]])

* Mutable Objects
: - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]])
: - Mutable Data Structures  ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]])

* IO Monad 
: - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]])
: - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]])  
: - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]])  

* ST Monad 
: - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]])
: - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]])
: - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]])

=== Monads IV : Reader and Writer Monads ===
* Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]])
* Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]])
* MonadState Class
:: - State &amp; StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]])
:: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]])
* MonadReader Class
:: - Reader &amp; ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]])
:: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]])
* Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]])

=== Monoid ===
* Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]])

=== Arrow ===
* Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]])
&lt;/br&gt;

==Polymorphism==
* Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]])
&lt;/br&gt;

==Concurrent Haskell ==
&lt;/br&gt;


go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://learnyouahaskell.com/introduction Learn you Haskell]
* [http://book.realworldhaskell.org/read/ Real World Haskell]
* [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material]


[[Category:Haskell|programming in plain view]]</text>
      <sha1>ms7ks3b5w7e4t7zpjbbwnx8166cyvfo</sha1>
    </revision>
    <revision>
      <id>2815087</id>
      <parentid>2815085</parentid>
      <timestamp>2026-06-10T19:07:11Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Lambda Calculus */</comment>
      <origin>2815087</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6273" sha1="lhaz6x9mu59f9q5l1el1mgmb71v3ydv" xml:space="preserve">
==Introduction==
* Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]])
* Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]])
* Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]])
* Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]])
* Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]])
&lt;/br&gt;

==Applications==
* Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]])
* Bird's Implementation 
:- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]])
:- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]])
:- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]])
:- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]])

&lt;/br&gt;

==Using GHCi==
* Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]])
&lt;/br&gt;

==Using Libraries==
* Library ([[Media:Library.1.A.20170605.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;


==Types==
* Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]])
* TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]])
* Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]])
* Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]])
* Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]])

==Functions==
* Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]])
* Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]])
* Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]])

==Expressions==
* Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]])
* Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]])
* Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;

==Lambda Calculus==
* Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]])
* Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]])
* Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]])
* Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]])
* Encoding Datatypes 
:- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]])
:- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]])
:- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]])
:- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]])
:- Recursions ([[Media:LCal.9A.Recursion.20260611.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]])
&lt;/br&gt;
&lt;/br&gt;

==Function Oriented Typeclasses== 
=== Functors ===
* Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]])
* Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]])
* Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]])

=== Applicatives ===
* Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]])
* Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]])
* Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]])
* Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]])

=== Monads I : Background ===
* Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]])
* Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]])
* Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]])
* Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]])
* IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]])
* Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]])

=== Monads II : State Transformer Monads ===
* State Transformer 
: - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]])
: - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]])
: - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]])

* State Monad 
: - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]])
: - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) 
: - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]])

=== Monads III : Mutable State Monads ===
* Mutability Background 
: - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]])
: - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]])
: - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]])
: - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]])
: - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]])

* Mutable Objects
: - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]])
: - Mutable Data Structures  ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]])

* IO Monad 
: - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]])
: - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]])  
: - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]])  

* ST Monad 
: - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]])
: - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]])
: - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]])

=== Monads IV : Reader and Writer Monads ===
* Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]])
* Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]])
* MonadState Class
:: - State &amp; StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]])
:: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]])
* MonadReader Class
:: - Reader &amp; ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]])
:: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]])
* Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]])

=== Monoid ===
* Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]])

=== Arrow ===
* Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]])
&lt;/br&gt;

==Polymorphism==
* Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]])
&lt;/br&gt;

==Concurrent Haskell ==
&lt;/br&gt;


go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://learnyouahaskell.com/introduction Learn you Haskell]
* [http://book.realworldhaskell.org/read/ Real World Haskell]
* [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material]


[[Category:Haskell|programming in plain view]]</text>
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    <title>Practicing Dialogue</title>
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        <username>Lbeaumont</username>
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      <comment>Improved links</comment>
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      <text bytes="38598" sha1="rq83wtra1zoidh7a3skfte7v50qy7tm" xml:space="preserve">—Thinking Together
==Introduction==
[[File:Kathy Matayoshi and Mazie Hirono.jpg|thumb|Two people practicing dialogue]]
{{TOC right | limit|limit=2}}
We have suddenly gone beyond ordinary conversation and are now beginning to listen, truly understand, learn from each other, and create together as we communicate [[Candor|candidly]]. We are thinking together, meaning now flows freely, and we are learning from the transformation that is [[w:dialogue|dialogue]].

'''Objectives'''

The objectives of this course are to:
*Recognize various forms of communication.
*Understand the benefits of using dialogue to communicate.
*Learn to use dialogue as your preferred method of communication.
*Experience a synthesis and interweaving of ideas.
*Gain insight as you dialogue with others.
{{100%done}}
Use this [[Practicing Dialogue/Daily Practicing Dialogue Checklist|daily practice checklist]] to make practicing dialogue a habit. {{By|lbeaumont}}
The course contains many [[w:Hyperlink|hyperlinks]] to further information. Use your judgment and these [[What Matters/link following guidelines|link following guidelines]] to decide when to follow a link, and when to skip over it.

This course is part of the [[Wisdom/Curriculum|Applied Wisdom curriculum]]. This material has been adapted from the EmotionalCompetency.com [http://emotionalcompetency.com/dialogue.htm page on dialogue], with permission of the author.

If you wish to contact the instructor, please [[Special:Emailuser/Lbeaumont | click here to send me an email]] or leave a comment or question on the [[Talk:Practicing_Dialogue|discussion page]].

[[File:Practicing Dialogue Audio Dialogue.wav|thumb|Practicing Dialogue Audio Dialogue]]

==Dialogue is Distinct==
[[w:Dialogue|Dialogue]] is the creative thinking together that can emerge when genuine [[w:Empathy|empathetic]] [[Knowing Someone/Deep Listening|listening]], respect for all participants, safety, [[w:Peer_group|peer]] relationships, suspending [[w:judgment|judgment]], sincere inquiry, courageous speech, and discovering and disclosing assumptions work together to guide our conversations. It is an activity of [[Fostering Curiosity|curiosity]], cooperation, creativity, discovery, and learning rather than persuasion, competition, fear, and conflict. Dialogue is the only [[w:Symmetry|symmetrical]] form of communication. Dialogue emerges from [[w:Trust_(social_sciences)|trusting]] relationships.

Dialogue is a form of conversation that is distinct from [[w:Conversation#Discussion|discussion]], [[w:debate|debate]], distraction, dismissal, delegation, disingenuous, diatribe, and [[w:dogma|dogma]] because dialogue is the only form of communication where the participants act as authentic peers. All other [[Communicating Power|forms of communication]] emphasize a [[w:Power_social_and_political|power relationship]] that interferes with the synthesis, analysis, and interweaving of ideas that characterize dialogue. Dialogue is driven by genuine [[Fostering Curiosity|curiosity]] and [[w:respect|respect]] rather than by power.  ''Deliberation'' describes a period of thought and reflection that can take place during any conversation. [[w:Rapport|Rapport]] is a close synonym to dialogue.
===Assignment===
*Listen to conversations and various other communications such as advertisement, advocacy, opinions, debates, etc.
*As you are [[Knowing Someone/Deep Listening|listening]], identify the [[Communicating Power|form of communication]] according to the power relationships being displayed. Name the form of communications as being: dialogue, discussion, debate, defense, distraction, dismissal, delegation, disingenuous, dialectic, decree, diatribe, or dogma.
* View the video [https://www.youtube.com/watch?v=0wnujWPW5U0 The solution is in the dialogue], by Peter Nixon presented April 2014 at TEDxHKUST 
*How often do you witness skillful dialogue?

==Toward Dialogue==
The goal of dialogue is ''insight'', the goal of [[w:argumentation|argumentation]] is often ''winning'' at the expense of insight. Specific attitudes, beliefs, and behaviors can move us toward dialogue or away from it, toward dichotomy and fragmentation. The following table characterizes the distinctions:

{| class=&quot;wikitable&quot;
|-
!Toward Dialogue !! Toward Dichotomy
|-
| Authentic curiosity, discovery, and disclosure. Revealing information, assumptions, and doubts. Done with others. [[w:I_and_Thou|I, thou]]. || Disingenuous manipulation, secrecy, and persuasion. Disguising and defending assumptions and doubts. Maintaining distance through a polite façade or direct confrontation. Done to others. I, it.
|-
| Cooperation and genuine respect. Peer relationships; equality. [[Earning Trust|Trust]] and safety. [[Candor]]. Willing collaborators. || Competition, criticism, and dismissal. Displaying power; coercion. Distrust and danger.
|-
| Insight.|| Insult.
|-
| Assume Positive Intent. || Combative, competitive, malicious intent, seeking revenge, getting even, retaliation.
|-
| [[W:Principle of charity|Principle of charity]]. [[w:Straw_man#Steelmanning|Steelmanning]]. || Attack, [[w:Gotcha_journalism|gotcha]], [[w:Straw_man|strawman]].
|-
| Listening to understand. Empathy. || Listening to respond and rebut; reloading. Apathy.
|-
| [[Clear_Thinking/Curriculum|Clear Thinking]]. || [[w:Rhetoric|Rhetorical Gamesmanship]].
|-
| [[Intellectual Honesty|Intellectual honesty]]. || [[w:Motivated_reasoning|Motivated reasoning]].
|-
| Exploring, examining, innovating, insight. Inquiry. 	||Making and scoring points. I win, you lose. Advocacy.
|-
| Choosing to explore; inventing new ideas, creating, learning, thinking. 	||Choosing to ignore; defending old postures, thoughts, and assumptions.
|-
| Scout mindset—Reasoning is like mapmaking. Decide what to believe by asking “Is this true?” Seek out [[Evaluating Evidence|evidence]] that will make your map more accurate.&lt;ref&gt;{{cite book |last=Galef |first=Julia |author-link=w:Julia_Galef|date=April 13, 2021 |title=The Scout Mindset: Why Some People See Things Clearly and Others Don't |publisherPortfolio  |pages=288 |isbn=978-0735217553}}&lt;/ref&gt;||Soldier mindset—Reasoning is like defensive combat. Decide what to believe by asking either “Can I believe this?” or “Must I believe this?” depending on your motives. Seek out evidence to fortify and defend your beliefs. 
|-
| [[Facing Facts/Reality is our common ground|Reality is our common ground]]. [[Seeking True Beliefs|Let's seek it together]].||Reality is what I say it is. Listen to me.
|-
| Abandoning reason is an act of violence. ||[[w:The_Art_of_Being_Right|Win at all costs]].
|-
|Synthesis, combination, alternative viewpoints, integration, coherence, new possibilities. Collective intelligence. Building up, feeling constructive. [[Finding Common Ground|Finding common ground]].	||Polarized, dichotomous thinking. Fragmentation and incoherence. Focusing on fears. Anxiety. Arrogance. Tearing down, feeling destructive.
|-
|[[w:Appreciative_inquiry|Appreciative inquiry]]. Shared inquiry. Seeking the strengths and possibilities in the other's ideas. Discernment. 	||Criticism. Searching for flaws and weakness in the other's ideas. Judgment.
|-
|Deferring closure to allow complete understanding, agreement, and enduring support. 	||Closing quickly to solidify your position.
|-
|[[Recognizing Fallacies|Identifying faulty reasoning]], information, inconsistencies, or assumptions. Willing to give up ground. 	||Attacking the person. Taking ground.
|-
|Seeking an inclusive viewpoint; valuing and accommodating diversity. Revealing assumptions and discrepancies. ||	Advocating a one-sided point of view; valuing conformance. Defending a point of view and the assumptions it encompasses.
|-
|I can learn from you. Inclusiveness. Our doubts help to cleanse our truths. 	||I am right, just listen to me. Be reasonable, do it my way. Resistance is futile.
|-
|[[Finding Courage|Courageous]] speech. [[Candor]]. 	||Serial monologue, harangue, attacks, bloviation, obfuscation, equivocation, posturing, rehashing, gossip, small talk, party line, and idle chatter.
|-
|Balance of advocacy and inquiry. 	||Advocacy displaces inquiry.
|-
|Comfortable with complexity and subtlety while seeking elegance. 	||Simplistic.
|-
|Together we can seek the truth. Let's journey together to find it. 	||I know the truth. It's my way or the highway.
|-
|Essence; a journey to the center of the being. Curiosity and flow. 	||Image. Fear, anxiety, and [[Resolving Anger|anger]].
|-
|Initial doubts leading to enduring certainty. 	||Initial certainty leading to enduring doubts.
|}

Dialogue is more subtle and cooperative than discussion or debate. However, as a minimum, participants in dialogue must adhere to the [[w:Pragma-dialectics#Rules_for_a_critical_discussion|rules for a critical discussion]]. 

Another minimum standard for dialogue are [[w:Rogerian_argument#Rapoport's_rules|Rapoport's Rules]], as restated here by Daniel Dennett:&lt;ref&gt;{{cite book |last=Dennett  |first=Daniel C. |date=May 5, 2014 |author-link=w:Daniel_Dennett |title=Intuition Pumps And Other Tools for Thinking |publisher=W. W. Norton &amp; Company |pages=512 |isbn=978-0393348781}} Chapter 3&lt;/ref&gt;
# You should attempt to re-express your target’s position so clearly, vividly, and fairly that your target says, “Thanks, I wish I’d thought of putting it that way.&quot; This is called [[w:Straw_man#Steelmanning|steelmanning]] the argument.
#You should list any points of agreement (especially if they are not matters of general or widespread agreement).
#You should mention anything you have learned from your target.
#Only then are you permitted to say so much as a word of rebuttal or criticism.

Yet another technique for discovering common ground is for each participant to answer these questions:
#What about your dialogue partner's position appeals to you?
#What about your own position troubles you?

Please keep in mind that in dialogue the only ''target'' is insight.
===Balance Inquiry and Advocacy===
[[File:Inquiry and advocacy.jpg|350px|right|The four essential skills of dialogue.]]

Dialogue requires the skillful use of four distinct practices to balance ''inquiry''—seeking to understand—and ''advocacy''—being understood. These can achieve the rhythm of respiration, first inhaling the ideas of others and later exhaling expression of your new ideas. These four skills: ''listen'', ''suspend'', ''respect'', and ''voice'' appear in the diagram on the right and are described more fully below.

'''Listening to understand:''' Hear their words; [[Knowing Someone/Deep Listening|learn their meaning]]. What is the person saying? What ideas do they want to get across? What are they feeling now? What is important to them? What does this mean for them? What is not being heard? Why? What is their truth? How can I connect with them? What can I learn from them? What have I been missing? What are we all missing? How can this new information change my point of view? Who is not being heard? What are the inconsistencies, dilemmas, and paradoxes? What new frame of reference can provide coherence? Concentrate on direct observation, stick to the facts, dismiss your old thoughts and assumptions, stay in their moment, hear their story, and defer interpretation. Listen without resistance as you notice your own resistance. Notice how you are reacting. Be still; stay silent inwardly and outwardly.

'''Suspending judgment:''' Defer your certainty while you explore doubt and new possibilities.  Stop, step back, adopt a new point of view, and reflect from this new vantage point. Frame up—adopt a broader reference frame. Allow inquiry to displace certainty. Embrace your ignorance. Be willing to disclose your own doubts. Acknowledge what you don't know and don't understand. What am I missing? What am I protecting? Reject polarized thinking. Hold your tongue and defer forming opinions, jumping to conclusions, quick fixes, and assigning [[Attributing Blame|blame]]. Become aware of your inner reaction, but don't react outwardly. Have the discipline to hold the tension within yourself while you silently examine and reflect on it. Remain curious. Identify and examine your assumptions and theirs. Work to understand how this problem works, how has it arisen? Cope constructively with your fears and [[Resolving Anger|anger]]. Do not attribute motive or intent. Don't yet agree or disagree while you remain curious and reflect. Defer and dismiss conclusions, explore alternative meanings and motives, integrate these new ideas with the whole, and seek congruence.

'''Respecting all:''' Attribute positive motives and constructive intent to each participant. Appreciate all that is good about them, all that you share in common with them, and all they can contribute. Acknowledge the dignity, legitimacy, worth, and humanity of the person speaking. Allow for differing viewpoints and learn all you can from them. Examine the origins within your self of any tendency you have to disrespect participants. Resist your temptation to [[Attributing Blame|blame]]. Remain humble and accept that they can teach us and we can learn from them. Attain and appreciate their viewpoint; do not attack, intrude, deny, dismiss, dispute, or discount their comments. Banish violence.

'''Speaking your voice:''' Contribute your insight to advance the dialogue. Be patient and gather your own clear thoughts before you speak with [[candor]]; clearly, directly, and authentically. What is most important to express now? Offer your insights. Share how you feel, what you don't know, and your own doubts and concerns. Speak courageously from your own authentic voice. Avoid sarcasm, barbs, attacks, insults, reification, and condescension. Inquire and ask only genuine questions arising out of [[Fostering Curiosity|curiosity]] and not belligerence. Test assumptions. Speak in the first person from your actual experiences. Speak your truth.

Dialogue is a dynamic process that requires a delicate balance. Inquiry—seeking new understanding—combines the skills of listening while suspending judgment to gain a deeper and newer understanding. This is balanced by advocacy—seeking to be understood. Advocacy combines respect for all participants with the courage to speak your voice, share your insights, and advance the dialogue toward a new understanding of the whole. Dialogue requires a balance between the analysis of inquiry and the action of advocacy. Inquiry and analysis alternate in balance with advocacy and action. The diagram illustrates a spiral path that encourages dialogue to emerge.  Beginning with listening, we then suspend and reflect, respect others, and then speak our voice before resuming our listening. The dialogue advances the group toward the whole at the center as the participants think together.

===Dynamic Roles===
Family therapist [[w:David_Kantor|David Kantor]] describes four distinct roles that dialogue participants adopt dynamically as the dialogue proceeds:
[[File:Dynamic roles during dialogue.jpg|250px|right|Dynamic roles during dialogue.]]
'''Move:''' Initiate action to move the dialogue in a particular direction. Set a direction and provide clarity.

'''Follow:''' Support, amplify, or derive a similar direction suggested by the preceding move.

'''Oppose:''' Raise an objection to highlight possible problems or point out what may not be quite right with the current direction.

'''Bystand:''' Propose a new way of thinking, a new viewpoint, a new reference frame, or a new direction that bypasses, transcends, or overcomes the temporary deadlock, expands the thinking of the group, and shows the way toward further progress. Provide perspective and encourage reflection.

All four roles are required to move the dialogue along. People fill one of the roles temporarily as the conversation needs each particular type of contribution to move forward. Each role takes into account the variety of viewpoints already expressed, incorporating much of the information that has been suspended during the dialogue. The roles are dynamic, the person who ''opposed'' in one instance may ''move'' in another or ''bystand'' later on. All four roles are necessary. Without a move, there is no direction. Without the follow there is no momentum. Without the opposition, there is no critical thinking and correction, and without the bystanders, deadlocks persist and there is no breakthrough to new understanding.

Conversation groups that do not achieve dialogue often get stuck in a move-oppose cycle that repeats without making progress.

The maze shown at the right illustrates how the four roles work together to move the group toward the shared central understanding; the whole at the center. The ''move'' gets thing started and the ''follow'' helps keep things going. However progress seems stalled when it encounters ''opposition''. After considering all viewpoints, the ''bystander'' suggests a novel path for the group to continue along.

==Matters of Fact==
[[w:Facts|Facts]] deserve a seat at the table during any dialogue. Therefore, it is important to carefully distinguish among: 1) matters of fact, 2) matters of preference, or 3) matters of controversy throughout each dialogue.

Statements can be classified as one of the following three types:&lt;ref&gt;{{cite book |last1=Paul |first1=Richard |last2=Elder |first2=Linda |date=December 5, 2014 |title=Thinker's Guide to the Art of Socratic Questioning (Thinker's Guide Library),  |publisher=Foundation for Critical Thinking |pages=134 |isbn=978-0944583319}} Three Kinds of Questions.&lt;/ref&gt;
#'''Matters of fact'''. These statements can be assessed and verified through the correct use of [[Evaluating Evidence|evidence gathering]], and reasoning. A correct statement can be made with conviction. These statements declare “what is” and careful researchers agree on the answer. Examples include: The boiling point of water is 100° Centigrade,  gold is denser than lead, and the movie ''Spotlight'' won Best Picture in 2016. Notice the use of “is” to convey certainty in these statements. Do not argue matters of fact, research them instead.
#'''Matters of taste, preference, or opinion'''. Any claim is acceptable here, because the statement depends only on the preferences of the person making it. Examples include: I feel that purple is the most beautiful color, I prefer chocolate ice-cream to vanilla ice-cream, and I believe that Rembrandt was a better artist than Picasso. Notice the use of “prefer”, “feel”, and “believe” to convey a personal preference. Do not argue matters of preference, enjoy them.
#'''Matters of controversy'''. Although these are not opinions, sincere experts often disagree on the best answer or the best course of action. These statements propose “what ought to be” or they ask about a topic that is not yet fully and carefully explored or researched. Examples include: I believe the most pressing problem facing the world today is the lack of clean safe drinking water for all people, I think the best approach to reducing gun violence is to require comprehensive background checks for all gun purchases, and I believe incarceration rates are too high in the US. Notice the use of “believe” and “think” to convey personal positions here. Learn more about matters of controversy by exploring them with dialogue and the [[Socratic Methods|Socratic Method]].
===Assignment===
#Read this essay on the [[Knowing How You Know/Height of the Eiffel Tower|Height of the Eiffel Tower]].
#Read over this list of [[Socratic Methods/questions to classify|questions to classify]]. 
#Identify at least five of these questions in each of the following classifications: 1) matters of fact, 2) matters of preference, or 3) matters of controversy,
#During dialogue notice the verbs used by you and your partner to convey degrees of certainly and conviction. These include: “is”, “prefer”, “feel”, “believe”, “think”, and others. Ensure the verb chosen corresponds to the degree of certainty and conviction of the statement being made. Address, explore, and correct and mismatches.
#If the dialogue encounters disagreement on matters of fact, agree to research the fact and come to an agreement on the fact before continuing the dialogue. Expect you and your dialogue partner to converge on matters of fact as a result of [[w:Consilience|consilience]]. If you are unable to converge on matters of fact, begin to explore the differences in your [[Knowing_How_You_Know#What_is_a_Theory_of_Knowledge.3F|theories of knowledge]] that may be leading you toward differing conclusions. 
#Complete the Wikiversity course on [[Finding Common Ground]]. Find common ground.
#Use agreements on matters of fact as a common basis for you and your dialogue partner to move the dialogue forward. 
#If the dialogue encounters disagreement on matters of taste, note the differences and continue the dialogue without requiring resolution of this disagreement. There is no correct resolution of matters of taste.
#Only matters of controversy are within the useful realm of dialogue. Direct the dialogue toward these matters of controversy to move toward insights and learning together.
#Complete the course on [[Facing Facts]].

==Obstacles==
Dialogue is easily spooked. There are many common obstacles that prevent dialogue from emerging. Removing sources of fear, suspending the exercise of power, eliminating external influences, removing distractions, and providing excellent communication conditions can all promote dialogue.
=== Fear ===
Fear prevents dialogue. People are often afraid to trust other participants, consider new ideas, and open up to the new possibilities that dialogue requires. People hold back and fail to participate fully and genuinely because of their fears. Suspending judgment is often an act of courage. Remaining open to new ideas; doubting, questioning, or abandoning beliefs you have held for many years, adopting a new viewpoint, releasing attachments, hearing someone for the first time, abandoning the [[w:Status quo|status quo]], thinking in a new way, allowing for change, acknowledging your old habits and beliefs, abandoning your stubbornness, admitting you don't know or don't understand, admitting you may have been wrong, exposing vulnerabilities, anticipating the ramifications and future consequences of new ideas and agreements, becoming [[w:Authenticity_(philosophy)|authentic]] rather than merely polite; and confronting assumptions, issues, and people, can all be scary. These obstacles require courage to overcome. Speaking truth to power and challenging the opinions and beliefs of others requires courage. Finding your voice requires courageous thinking. Speaking your voice requires courageous action. Have the courage to dialogue.

Notice the relative salience of ''[[Fostering Curiosity|curiosity]]'' and ''fear''. Seek to attain and sustain curiosity. Identify and remove obstacles to curiosity. Notice when fear arises and displaces curiosity. Pause the conversation to note the emergence of fear, identify its causes, resolve the issues motivating the fear, and return to curiosity and dialogue.

=== External Constraints ===

Dialogue requires [[w:autonomy|autonomy]]. Speaking your voice requires thinking for yourself and making your own decisions. Dialogue requires adopting an internal [[w:Locus_of_control|locus of control]] and rejecting an external locus of control. Repeating the opinion of others, deferring your own judgment to someone outside the room, appealing to the views of your chosen experts or luminaries, defending a special interest, holding conflicting interests, running a secret agenda, reciting dogma, remaining star struck, going along to get along, deferring to fate or luck, or introducing external constraints such as “my boss requires . . .” or “everybody knows. . .” all prevent you from making your own decisions and speaking your own voice. Shed these external constraints so you can think for yourself, represent yourself, speak for yourself, and participate in the dialogue. Speak in the first person about your own experiences, [[w:Opinion|opinions]], and [[w:belief|beliefs]].

=== Distractions ===

Dialogue requires focus. Multitasking seems to be emerging as the new status symbol. But dialogue is hard work that requires your full and present attention. Listening for meaning requires focus and full attention. Suspending judgment requires self discipline. Speaking your voice requires presence and thoughtfulness. Respect often requires patience and cannot be rushed. Reading mail, talking on the phone, text messaging, surfing the net, side conversations, watching the clock, preparing for the next meeting, writing notes, showboating, or wishing you were elsewhere are all distractions that will prevent you from fully participating in dialogue. Your lack of attention and concern also distracts others and may prevent them from participating in dialogue. Either focus your full and undivided attention on the conversation, or leave the room. Expect this focus of the others.
=== Poor Communications ===

Dialogue requires careful, detailed, delicate, and nuanced communications. Poor room acoustics, physical distance, language differences, accents, jargon, local vernacular, unfamiliar vocabulary, cultural differences, unshared abstractions, [[w:logical fallacies|logical fallacies]], intentional and unintentional [[w:Cognitive_distortion|distortions]], hearing difficulties, and poor sound systems can all prevent dialogue from emerging. Collocated participants in a private room free of distractions sitting comfortably in a circle where everyone can easily see and hear everyone else promotes communication that can help dialogue emerge. If language differences exist, then effective translation services, including cross-cultural translations, are required.

=== Bad faith actors ===
Practicing dialogue requires both parties to act in good faith. Work to ensure that each participant intends to do their best to remain intellectually honest and abide by the dialogue guidelines described above. However, it is difficult to predict another’s behavior until the dialogue is underway, and you may encounter tricks used in bad faith to undermine the integrity of the  dialogue in an attempt to “win the argument”. One trick is called the [[w:gish gallop|Gish gallop]]. This is a rhetorical technique in which a person in a debate attempts to overwhelm their opponent by providing an excessive number of arguments with no regard for the accuracy or strength of those arguments.

[[w:Mehdi_Hasan|Mehdi Hasan]] suggests using these three steps to beat the Gish gallop: &lt;ref&gt;Mehdi Hasan, [https://cafe.com/stay-tuned/debating-101-with-mehdi-hasan/ Stay Tuned with Preet, Debating 101], March 16, 2023. &lt;/ref&gt;

#Because there are too many falsehoods to address, it is wise to choose one as an example. Choose the one weakest, dumbest, most ludicrous argument that the opponent has presented and tear this one argument to shreds. This is called the ''weak point rebuttal''.
#Don’t budge from ''this'' issue. Don’t move on from this issue until you have destroyed the nonsense and clearly and decisively made your point.
#Call it out, name the strategy. “This is a strategy called the ‘Gish Gallop’, don’t be fooled by the flood of nonsense you have just heard.”

Avoid entering into a debate with someone who is likely to use the Gish gallop.

=== Getting Unstuck ===
Even skillful dialogue can get stuck when It enters areas of high conflict. Journalist [[w:Amanda_Ripley|Amanda Ripley]] recommends “Complicating the Narratives”&lt;ref&gt;  [https://thewholestory.solutionsjournalism.org/complicating-the-narratives-b91ea06ddf63 Complicating the Narratives], June 27, 2018, Solutions Journalism, Amanda Ripley.&lt;/ref&gt; to add complexity, encourage a range of emotions to flow, allow [[Fostering Curiosity|curiosity]] to displace fear, invite the conversation to go deeper, and continue the dialogue.

She suggests continuing the dialogue by asking one or more of these questions:
*What is oversimplified about this issue?
*How has this conflict affected your life?
*What do you think the other side wants?
*What’s the question nobody is asking?
*What do you and your supporters need to learn about the other side in order to understand them better?
*Tell me more.
=== Counterfactual Questions ===
Many beliefs, biases, and unkind [[w:Stereotype|stereotypes]], although strongly held, are formed arbitrarily. The arbitrary basis for many such beliefs can be exposed, and often [[w:Belief_revision|revised]] or dislodged, by posing [[w:Counterfactual_conditional|counterfactual]] questions.&lt;ref&gt;{{cite book |last=Grant |first=Adam |author-link=w:Adam_Grant |date= |title=Think Again: The Power of Knowing What You Don't Know|publisher=Viking|pages=320 |isbn=978-1984878106}}, Chapter 6.&lt;/ref&gt;

For example, during dialogue ask questions like these as appropriate:

*If you were born in a different place, would you still hold that belief?
**If you grew up in New York City instead of Boston, would you be a [[w:Yankees–Red_Sox_rivalry|Yankee’s fan rather than a Red Sox fan]]?
**If you were born in a different country, would you practice a different religion?
***How would your beliefs differ?
**If you were born in a different country, would your patriotism be to another country?
***Would you be fighting on the other side of this war?
*Would you hold different beliefs if you:
**grew up poor;
**were born [[w:Disability|disabled]];
**were born another [[w:Race_(human_categorization)|race]];
**were born another sex;
**were [[w:Obesity|obese]];
**were unattractive; 
**grew up on a farm rather than in the city;
**chose different friends;
**had different parents;
**were an [[w:Orphan|orphan]];
**became fatally ill;
**spoke a different language;
**won the [[w:Lottery|lottery]];
**lived 1,000 years ago; or
**worked in a different [[w:Job|job]]?

How would your beliefs be different?

Why would your beliefs be different?

What is the basis for your current beliefs?

If you can get people to pause and reflect on the origins of their beliefs, they might decide that it is irrational to apply group stereotypes to individuals, or to hold strongly to arbitrarily formed beliefs.

=== Recovering Dialogue ===
Because dialogue is fragile, conversations that begin as dialogue can shift [[Communicating Power|their tone]] and become argumentative. Take care to notice when this happens, pause the conversation to announce this shift and restore the dialogue. 

The shift often occurs when [[Fostering Curiosity|curiosity]] yields to fear.

Throughout each dialogue session:
#Periodically notice the [[Communicating Power|tone of the conversation]] to determine if the rules of dialogue are being followed.
#If you notice a slip away from dialogue toward debate or other argumentative forms of communication, interrupt and pause the conversation. A request such as “Let’s take a break here to examine our dialogue skills in action” may provide an effective transition. 
#Describe the shift in conversational tone that you have noticed. It may be helpful to identify [[Practicing_Dialogue#Toward_Dialogue|specific dialogue characteristics]] that are missing, or the argumentative characteristics that have appeared. This conversation may be contentious. Cite specific examples such as “When you said X, were you speaking from curiosity or from power? (pause here to allow a thoughtful answer) Were you seeking insight or trying to win an argument? (pause) Were you assuming my positive intent, or becoming combative?”  
#Ask that these argumentative behaviors be abandoned. Resolve to return to the goal of ''insight'' rather than ''winning''. Allow this topic to be discussed if this helps identify the characteristics that distinguish dialogue from argumentation. It may be helpful to explore the prevailing [[Socratic_Methods#Essential_Socratic_Temperament|temperaments of the participants]] to determine if conditions are suitable for continuing dialogue. 
#It may be helpful to take a short break to reflect on what happened, improve your dialogue skills, strengthen your relationship, reaffirm your intent to continue dialogue, and sharpen your ability to notice when and how the dialogue transformed. 
#Resume the dialogue from some place before it became argumentative.
#Continue to monitor the tone of communication and pause if argumentation arises again.
#Use these [[/Phrases for managing conversations/]] throughout the process.

== Success Stories ==
The power of dialogue has achieved some successful solutions to very difficult problems. Here are some examples:

* The [http://compact.org/resource-posts/the-san-diego-dialogue/ San Diego Dialogue project] is contributing to the advancement of research, relationships and solutions to the San Diego-Baja California crossborder region's long-term challenges in innovation, economy, health and education.
* The [http://ncdd.org/ National Coalition for Dialogue and Deliberation] works to give people a voice in important issues. Their website documents many successful dialogue projects.
* [[w:Vicki_Robin#Conversation_Caf.C3.A9|Conversation Café]] groups are improving conversations and strengthening the interconnections among people across America.
*The [http://www.civilconversationsproject.org/ Civil Conversations Project] seeks to renew common life in a fractured and tender world.
* [https://livingroomconversations.org Living Room Conversations] are a conversational bridge across issues that divide and separate us.

==Assignment==
# Learn the [[Practicing_Dialogue#Toward_Dialogue|distinctive skills of dialogue]].
## Perform the exercises in this [[/Daily Practicing Dialogue Checklist/|daily practicing dialogue checklist]].
# Remove the [[Practicing_Dialogue#Obstacles|obstacles described above]].
# Assemble and engage the stakeholders.
# Create the space, increase safety, [[Earning Trust|build trust]], level power, defer decision making, demonstrate empathy.
# Invite the group to do something truly important, and then
# stand back.

Allow an important dialogue to emerge as meaning begins to flow.

==Optional Assignment==
# Read the essay [[/From Demagoguery to Dialogue/]]
# Stay alert for opportunities to intervene in a discussion gone bad, stop the action, ask for fact checking, remind the discussants of the rules for dialogue, and encourage the discussants to practice dialogue.

== Further Reading ==

Students interested in learning more about dialogue may be interested in the following materials:
*{{cite book |last=Yankelovich |first=Daniel |date=September 5, 2001 |title=The Magic of Dialogue: Transforming Conflict into Cooperation |publisher= Touchstone |pages=240 |isbn=978-0684865669}}
*{{cite book |last=Isaacs |first=William |date=September 14, 1999 |title=Dialogue: The Art Of Thinking Together |publisher=Crown Business |pages=448 |isbn=978-0385479998}}
*{{cite book |last=Bohm |first=David |date= |title=On Dialogue |publisher=Routledge |pages=144 |isbn=978-0415336413}}
*{{cite book |last=de Bono |first=Edward |date=August 18, 1999 |title=[[w:Six_Thinking_Hats|Six Thinking Hats]]   |publisher=Back Bay Books |pages=192 |isbn=978-0316178310}}  
*{{cite book |last=Runion |first=Meryl |date=December 31, 2003 |title=How to Use Power Phrases to Say What You Mean, Mean What You Say, &amp; Get What You Want |publisher=McGraw-Hill Education |pages=224 |isbn=}}
*{{cite book |last1=Fisher |first1=Roger |last2=Ury  |first2=William L. |date=December 1, 1991 |title=[[w:Getting_to_Yes|Getting to Yes]]: Negotiating Agreement Without Giving In |publisher=Penguin Books |pages=200 |isbn=978-0140157352}}   
*{{cite book |last=Ury |first=William |date=February 27, 2007 |title=The Power of a Positive No: How to Say No and Still Get to Yes |publisher=Bantam |pages=272 |isbn=978-0553804980}}
*{{cite book |last1=Fisher |first1=Roger |last2=Shapiro |first2=Daniel |date=October 6, 2005 |title=Beyond Reason: Using Emotions as You Negotiate |publisher=Viking Adult |pages=256 |isbn=978-0670034505}}
*{{cite book |last1=Miller |first1=William R. |last2=Rollnick |first2=Stephen |date=April 12, 2002 |title=[[w:Motivational_interviewing|Motivational Interviewing]]: Preparing People for Change, 2nd Edition |publisher=The Guilford Press |pages=428 |isbn=978-1572305632}}
*{{cite book |last=Frankfurt |first=Harry G. |date=January 30, 2005 |title=[[w:On_Bullshit|On Bullshit]] |publisher=Princeton University Press |pages=67 |isbn= 978-0691122946}}
*{{cite book |last=Galtung |first=Johan |date=July 1, 2004 |title=Transcend and Transform: An Introduction to Conflict Work |publisher=Routledge |pages=200 |isbn=978-1594510632}}
*{{cite book |last=Friedman |first=Maurice S. |date=November 10, 2002 |title=Martin Buber: The Life of Dialogue |publisher=Routledge |pages=432 |isbn=978-0415284752}}
*{{cite book |last=Galef |first=Julia |author-link=w:Julia_Galef|date=April 13, 2021 |title=The Scout Mindset: Why Some People See Things Clearly and Others Don't |publisherPortfolio  |pages=288 |isbn=978-0735217553}}
*{{cite book |last=Briskin |first=Alan |date=October 1, 2009 |title=The Power of Collective Wisdom: And the Trap of Collective Folly |publisher=Berrett-Koehler Publishers |pages=220 |isbn=978-1576754450}}
*{{cite book |last=Grant |first=Adam |author-link=w:Adam_Grant |date= |title=Think Again: The Power of Knowing What You Don't Know|publisher=Viking|pages=320 |isbn=978-1984878106}}
* [https://www.ted.com/talks/robb_willer_how_to_have_better_political_conversations How to have better political conversations], TED Talk, September 2016, Robb Willer
* [http://www.npr.org/programs/ted-radio-hour/558307433 Dialogue And Exchange], TED Radio Hour program, Friday October 27, 2017
* [https://www.ted.com/talks/joan_blades_and_john_gable_free_yourself_from_your_filter_bubbles?language=en Free yourself from your filter bubbles], TED Talk, November 2017, Joan Blades and John Gable

I have not yet read the following books, but they seem interesting and relevant. They are listed here to invite further research.
*[[w:The_Art_of_Being_Right|''The Art of Always Being Right'']]: ''Thirty Eight Ways to Win When You Are Defeated'', by Grayling, A. C.

==References==
&lt;references/&gt;
{{Emotional Competency}}
{{CourseCat}} 
[[Category:Life skills]]
[[Category:Applied Wisdom]]
[[Category:Philosophy]]
[[Category:Peace studies]]
[[Category:Humanities courses]]
[[Category:Community]]
[[Category:Social Skills]]</text>
      <sha1>rq83wtra1zoidh7a3skfte7v50qy7tm</sha1>
    </revision>
  </page>
  <page>
    <title>User:ThaniosAkro/sandbox</title>
    <ns>2</ns>
    <id>219126</id>
    <revision>
      <id>2815176</id>
      <parentid>2780875</parentid>
      <timestamp>2026-06-11T08:34:55Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Table */</comment>
      <origin>2815176</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="55992" sha1="s49j76f66cdfrh8zjsxckym2e2dqj3q" xml:space="preserve">

==Table==

{| class=&quot;wikitable&quot;
|-
| From &lt;math&gt;(3):&lt;/math&gt; || &lt;math&gt;p^3&lt;/math&gt;|| || &lt;math&gt;+Ap&lt;/math&gt;  || &lt;math&gt;+B&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (5)&lt;/math&gt;
|-
| From &lt;math&gt;(4):&lt;/math&gt; || || &lt;math&gt;Cp^2&lt;/math&gt;|| &lt;math&gt;&lt;/math&gt;  || &lt;math&gt;+D&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (6)&lt;/math&gt;
|-
| &lt;math&gt;(5)*D&lt;/math&gt; || &lt;math&gt;Dp^3&lt;/math&gt;|| || &lt;math&gt;+DAp&lt;/math&gt;  || &lt;math&gt;+DB&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (7)&lt;/math&gt;
|-
| &lt;math&gt;(6)*B&lt;/math&gt; || || &lt;math&gt;BCp^2&lt;/math&gt;|| &lt;math&gt;&lt;/math&gt;  || &lt;math&gt;+BD&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (8)&lt;/math&gt;
|-
| &lt;math&gt;(7)-(8)&lt;/math&gt; || &lt;math&gt;Dp^3&lt;/math&gt;||&lt;math&gt;-BCp^2&lt;/math&gt; || &lt;math&gt;+DAp&lt;/math&gt;  || &lt;math&gt;&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (9)&lt;/math&gt;
|-
| Simplify &lt;math&gt;(9)&lt;/math&gt; || &lt;math&gt;&lt;/math&gt;||&lt;math&gt;Dp^2&lt;/math&gt; || &lt;math&gt;-BCp&lt;/math&gt;  || &lt;math&gt;+DA&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (10)&lt;/math&gt;
|-
| &lt;math&gt;(6)*A&lt;/math&gt; || || &lt;math&gt;ACp^2&lt;/math&gt;|| &lt;math&gt;&lt;/math&gt;  || &lt;math&gt;+AD&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (11)&lt;/math&gt;
|-
| &lt;math&gt;(10)-(11)&lt;/math&gt; || &lt;math&gt;&lt;/math&gt;||&lt;math&gt;Dp^2-ACp^2&lt;/math&gt; || &lt;math&gt;-BCp&lt;/math&gt;  || &lt;math&gt;&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (12)&lt;/math&gt;
|-
| Simplify &lt;math&gt;(12)&lt;/math&gt; || &lt;math&gt;&lt;/math&gt;|| || &lt;math&gt;Dp-ACp&lt;/math&gt; || &lt;math&gt;-BC&lt;/math&gt;  || &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (13)&lt;/math&gt;
|}

==Complex Cube Root==

See [https://en.wikiversity.org/wiki/Cubic_function#Vieta's_substitution Vieta's Substitution.]

The depressed cubic function is : &lt;math&gt;f(t) = t^3 + At + B = 0.&lt;/math&gt;

Let &lt;math&gt;A = -3C&lt;/math&gt; and let &lt;math&gt;t = w + \frac{C}{w} = \frac{w^2 + C}{w}.&lt;/math&gt;

Substitute for &lt;math&gt;A, t&lt;/math&gt; in the depressed function and result is:

&lt;math&gt;f(w) = w^6 + Bw^3 + C^3&lt;/math&gt; or

&lt;math&gt;f(W) = W^2 + BW + C^3&lt;/math&gt; where &lt;math&gt;W = w^3&lt;/math&gt; and &lt;math&gt;w = \sqrt[3]{W}&lt;/math&gt;.

From the quadratic formula: &lt;math&gt;W = \frac{-B \pm \sqrt{B^2 - 4C^3}}{2} = \frac{-B \pm \sqrt{4C^3 - B^2}\sqrt{-1}   }{2}&lt;/math&gt;

However, &lt;math&gt;W = a + bi.&lt;/math&gt;

Therefore: &lt;math&gt;a = \frac{-B}{2}&lt;/math&gt; and &lt;math&gt;b = \frac{ \sqrt{4C^3 - B^2} }{2}.&lt;/math&gt;

&lt;math&gt;B = -2a&lt;/math&gt; and

&lt;math&gt;2b = \sqrt{4C^3 - B^2} &lt;/math&gt;

&lt;math&gt;4b^2 = 4C^3 - B^2 &lt;/math&gt;

&lt;math&gt;4C^3 = 4b^2 + B^2 &lt;/math&gt;

&lt;math&gt;C^3 = b^2 + \frac{B^2}{4} &lt;/math&gt;

&lt;math&gt;C = \sqrt[3]{C^3} &lt;/math&gt; and

&lt;math&gt;A = -3C. &lt;/math&gt;

&lt;math&gt;A,B &lt;/math&gt; of &lt;math&gt;f(t)&lt;/math&gt; have been defined with &lt;math&gt;C&lt;/math&gt; positive and &lt;math&gt;A&lt;/math&gt; negative.

Because &lt;math&gt;W&lt;/math&gt; has three complex roots, &lt;math&gt;f(t)&lt;/math&gt; must have three real roots.

Calculate one of the roots of &lt;math&gt;f(t).&lt;/math&gt;

&lt;math&gt;t = \frac{w^2 + C}{w}.&lt;/math&gt;

Therefore &lt;math&gt;wt = w^2 + C&lt;/math&gt; and

&lt;math&gt;w^2 - tw + C = 0.&lt;/math&gt;

From the quadratic formula : 

&lt;math&gt;w = \frac{t \pm \sqrt{t^2 - 4C}}{2} = \frac{t \pm \sqrt{4C - t^2}\sqrt{-1}   }{2}&lt;/math&gt;

&lt;math&gt;w = p + qi&lt;/math&gt;

Therefore: &lt;math&gt;p = \frac{t}{2}&lt;/math&gt; and &lt;math&gt;Q = \frac{4C - t^2  }{4} = C - \frac{t^2}{4}&lt;/math&gt; where &lt;math&gt;Q = q^2.&lt;/math&gt;

&lt;math&gt;q = \sqrt{Q}&lt;/math&gt; but sign of this calculation of &lt;math&gt;q&lt;/math&gt; is ambiguous.

Let &lt;math&gt;P = p^2&lt;/math&gt;

&lt;math&gt;q = \frac{b}{(3P - Q)}&lt;/math&gt;

and &lt;math&gt;(p + qi)^3 = a + bi.&lt;/math&gt;
===An Example===

Calculate cube roots of complex number &lt;math&gt;W = -39582 + 3799i.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:1215depressed cubic01.png|thumb|400px|'''Graph of &lt;math&gt;f(t)&lt;/math&gt; shown as graph of &lt;math&gt;f(x)&lt;/math&gt; and showing three values of &lt;math&gt;t: t_1, t_2, t_3&lt;/math&gt;.'''
&lt;/br&gt;
&lt;math&gt;Y&lt;/math&gt; axis compressed for clarity.
&lt;/br&gt;
&lt;math&gt;A,B = -3495.0, 79164.0&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;t_1 = -68.229473419497441512295943903670298\dots&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;t_2 = 32.229473419497441512295943903670298\dots&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\ t_3 = 36&lt;/math&gt;
]]

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

import decimal
decimal.getcontext().prec = 22
ab = -39582,3799
a,b = [ dD(v) for v in ab ]
B = -2*a
C = (b**2 + B*B/4) ** (dD(1)/3)
A = -3*C

a,b,A,B,C = [ dD(str(float(v))) for v in (a,b,A,B,C) ]
print ( 'a,b = {}, {}'.format(a,b) )
print ( 'A,B,C = {}, {}, {}'.format(A,B,C) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
a,b = -39582.0, 3799.0
A,B,C = -3495.0, 79164.0, 1165.0
&lt;/syntaxhighlight&gt;
Calculate roots of cubic function:

&lt;math&gt;y = f (t) &lt;/math&gt;&lt;math&gt; = t^3 &lt;/math&gt;&lt;math&gt; - 3495 t &lt;/math&gt;&lt;math&gt; + 79164 .&lt;/math&gt;

Three roots are: &lt;math&gt;t_1,\ t_2,\ t_3 = -68.22947341949744\dots,\ 32.22947341949744\dots,\ 36.0&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
# python code:
t1, t2, t3 = ('-68.229473419497441512295943903670298',
               '32.229473419497441512295943903670298', 36 )
values_of_t = [ dD(v) for v in (t1,t2,t3) ]

for t in values_of_t :
    p = t/2 ; P = p**2
    Q = C - t*t/4 ; q = b/(3*P - Q)
    # Check results:
    print ()
    sx = 't' ; print (sx,'=', eval(sx))
    print ( 'p,q = {}, {}'.format(p,q) )
    ab = [ p*P - 3*p*Q, 3*P*q - q*Q ]
    a_,b_ = [ float(v) for v in ab ]
    sx = 'a_,b_' ; print (sx,'=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
t = -68.229473419497441512295943903670298
p,q = -34.11473670974872075615, 1.088457268119895641747
a_,b_ = (-39582.0, 3799.0)
    
t = 32.229473419497441512295943903670298
p,q = 16.11473670974872075615, -30.08845726811989564176
a_,b_ = (-39582.0, 3799.0)
    
t = 36
p,q = 18, 29
a_,b_ = (-39582.0, 3799.0)
&lt;/syntaxhighlight&gt;
Three cube roots of &lt;math&gt;W = -39582 + 3799i&lt;/math&gt; are:

&lt;math&gt;w_1 = -34.11473670974872075615 + 1.088457268119895641747i&lt;/math&gt;

&lt;math&gt;w_2 = 16.11473670974872075615 - 30.08845726811989564176i&lt;/math&gt;

&lt;math&gt;w_3 = 18 + 29i&lt;/math&gt;

==Making the Decimal object==

The following function verifies that we are working with Decimal objects. 

&lt;syntaxhighlight lang=python&gt;
import sys
import decimal

getcontext = decimal.getcontext
dD = D = Decimal = decimal.Decimal
DT = decimal.DecimalTuple
dgt = decimal.getcontext()

dgt.prec = 50


def print_error (error, x=None, thisName=None) :
    &quot;&quot;&quot;
Prints error derived from sys.exc_info().
    &quot;&quot;&quot;
    list1 = error.split(',')
    v1 = ','.join(list1[1:-1])
    if thisName : print (thisName)
    if x != None : print ('  Input =', (str(x))[:60])
    print (' ', list1[0])
    print (' ', v1)
    print (' ', list1[-1])


def makeDecimal (x, flag = 0) :
    '''
output = makeDecimal (x [, flag]) 
x is a single object convertible to Decimal object.
returns Decimal object or
returns None on error.
'''
    thisName = 'makeDecimal (x) :'
    if flag :
        print ()
        print (thisName)
        print ('  Input x =', type(x), (str(x))[:60])
        
    if isinstance(x, dD) : return x
    if isinstance(x, int) : return dD(x)
    if isinstance(x, float) : return dD(str(x))
    if isinstance(x, str) :
        try :
            error = ''
            output = dD(x)
        except : error = str(sys.exc_info())
        if error :
            if flag : print_error (error) 
            return None
        return output
        
    if isinstance(x, (list,tuple,DT)) :
        try :
            error = ''
            v1,v2,v3 = x
            if isinstance(v2, (tuple,list)) : output = dD(x)
            else : output = dD( ( v1, list(v2), v3 ) )
        except : error = str(sys.exc_info())
        if error :
            if flag : print_error (error) 
            return None
        return output

    if flag :
        print ('  Input not recognized.')
    return None
&lt;/syntaxhighlight&gt;
===checkComplex(x)===

This function verifies that the object is a valid complex tuple.

&lt;syntaxhighlight lang=python&gt;
def checkComplex(x, flag = 0) :
    &quot;&quot;&quot;
status = checkComplex(x [, flag]) 
x must be :
    (v1,v2,'rectangular') or
    (v1,v2,'polar')
    v1,v2 must be type Decimal.
    &quot;&quot;&quot;
    thisName = 'checkComplex (x) :'

    if flag :
        print ()
        print (thisName)
        print ('  Input x =', type(x), (str(x))[:60])

    try:
        error = ''
        v1,v2,str1 = x
        isinstance (v1,dD) or ({}['v1 must be type Decimal.'])
        isinstance (v2,dD) or ({}['v2 must be type Decimal.'])
        (isinstance (str1,str) and str1) or ({}['str1 must be valid string.'])
        str1 = str1.lower()
        if str1 == 'rectangular'[:len(str1)] : pass
        elif str1 == 'polar'[:len(str1)] : pass
        else : ({}['str1 must be &quot;rectangular&quot; or &quot;polar&quot;.'])
    except : error = str(sys.exc_info())
    if error :
        if flag : print_error (error)
        return False
    return True
&lt;/syntaxhighlight&gt;

===str_to_complex (input_string)===

&lt;syntaxhighlight lang=python&gt;
import re

# The re pattern for numbers to be found in string of type complex.
# In string like '  1234567890987654323456E-6 + 987654321234567890987e+2J   '
# extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'.
# eval allows '  12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J   '.
# Conversion to Decimal allows: '__987_65432123____4567890_987___'.
# eval does not allow ' 0123 ' but allows '  2e04 '.
# Conversion to Decimal allows: ' 0123  '
digits  = '[_0123456789]'
integer = '[_0123456789]{1,}'                    # Matches 1 23 345 4567 56789
exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8
Exponent = '({}){{0,1}}'.format(exponent)        # exponent 0 or 1 times.
float1 = '({})\.{{0,1}}'.format(integer)         # Matches 123  2345.
float2 = '{}{{0,}}\.{}'.format(digits, integer)  # Matches   123.345  .345
float_ = '(({})|({}))'.format(float2,float1)
rvalue = '{}({})'.format(float_,Exponent)
rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue)
ivalue_signed = '{}[Jj]'.format(rvalue_signed)


def str_to_complex (input_string) :
    &quot;&quot;&quot;
a,b = str_to_complex (input_string)
input_string could be '   1234567890987654323456E-6 + 987654321234567890987e+2J    '
input_string could be ' ' '  ( 12345678909876543234561234567890987.654323456E-6 +
                             987654321234567890987987654321.23764567890987e+2J  )  ' ' '
or ' ' '
    - (
 (
    12345678_90543123456789054_312345678905431234567890543e-014 +
    543123456789054_3123456789054312345_67890543654327453E-1_3J
) )

    ' ' '
or ' ' '
    - (
 (
    12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_'
    543123456789054_3123456789054312345_67890543654327453E-1_3J
) )

    ' ' '
or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)'
or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)'
or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)',
This function retains precision of a,b
    &quot;&quot;&quot;
    
    try :
        status = 0
        isinstance(input_string,str) or ({}['input_string not type str.'])
        str1 = input_string.strip()
        cx1 = eval(str1)
        isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.'])
    except : status = 1
    if status : return

    # This code removes white lines and comments, if any, at end of each line.
    new_line = '''
'''[-1:]
    lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ]
    lines = [ v for line in lines
                for parts in [ line.split('#') ]
                for v in [ parts[0]] ]
    str1 = ''.join(lines)
    str1 = ''.join(str1.split())

    if isinstance (cx1, (int,float)) :
        resultr = re.search(rvalue_signed, str1)
        resultr or ({}['rvalue not found.'])
        dD1 = dD(resultr[0])
        v1 = eval(str(dD1))
        if v1 == cx1 : return dD1,dD(0)
        if v1 == -cx1 : return -dD1,dD(0)
        ({}['dD1 not recognized.'])

    # cx1 must be complex. It must contain imaginary value.
    resulti = re.search(ivalue_signed, str1)
    resulti or ({}['ivalue not found.'])
    str2j = resulti[0] ; str2 = str2j[:-1]
    dD2 = dD(str2)
    str1 = ''.join(str1.split(str2j))

    # cx1 may contain real value.
    resultr = re.search(rvalue_signed, str1)
    if resultr : dD1 = dD(resultr[0])
    else : dD1 = dD(0)

    cx2 = complex(dD1,dD2)
    if cx2.real == cx1.real : pass
    elif cx2.real == -cx1.real : dD1 = dD1.copy_negate()
    else : ({}['dD1 Not Recognized.'])
    if cx2.imag == cx1.imag : pass
    elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate()
    else : ({}['dD2 Not Recognized.'])

    cx3 = complex(dD1, dD2)
    if cx3 == cx1 : return dD1,dD2
    ({}['No match for cx3.'])

&lt;/syntaxhighlight&gt;

===makeComplex(x)===

&lt;syntaxhighlight lang=python&gt;
def makeComplex (x, flag = 0) :
    '''
result = makeComplex (x[, flag]) 

Input can be tuple with 1,2 or 3 members.
If 1 or 2 members, 'rect' is understood.
The one member or single object may be int, float, complex, CompleX
or string convertible to int, float or complex.

x = makeComplex(4)
x = makeComplex((4,))
x = makeComplex(('4',0))
x = makeComplex((4,'0', 'rect'))
x = makeComplex(4+0j)
x = makeComplex('4+0j')
x = makeComplex(('4+0j',))
In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' )

output is always
(modulus, phase, &quot;polar&quot;) or
(real_part, imag_part, &quot;rect&quot;)
modulus, phase, real_part, imag_part are Decimal objects.

On error returns None.
'''

    thisName = 'makeComplex (x) :'

    if flag :
        print ()
        print (thisName)
        print ('  Input x =', type(x), (str(x))[:60])

    if isinstance(x, CompleX) :  # New class CompleX (note the punctuation.)
        x.check()
        return (x.r, x.i, 'rect')
    
    if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) 

    try :
        status = 1
        a,b = str_to_complex (x) 
    except : status = 0
    if status : return a,b,'rect'

    try :
        status = 1
        result = makeDecimal(x)
        isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.'])
        output = result,dD(0),'rect'
    except : status = 0
    if status : return output

    try :
        status = 1
        v1, = x # Allow for (( 3+4j ),)
    except : status = 0
    if status : return makeComplex (v1)

    try :
        status = 1 ; error = ''
        if len(x) == 2 :
            v1,v2 = [ makeDecimal (v) for v in x  ]
            str1 = 'rect'
        elif len(x) == 3 :
            v1,v2,str1 = x
            v1,v2 = [ makeDecimal (v) for v in (v1,v2)  ]
        else : ({}['len(x) not in (2,3)'])
        output = v1,v2,str1
        checkComplex(output) or ({}['output not valid complex.'])
        if str1[0] in 'rR' : output = v1,v2,'rect'
        else : output = v1,v2,'polar'
    except : status = 0 ; error = str(sys.exc_info())
    if status : return output
    
    if flag : print_error (error)
    return 
&lt;/syntaxhighlight&gt;

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

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of
the section, and also how to calculate the foci and directrices given the equation.
==Latera recta et cetera==
&quot;Latus rectum&quot; is a Latin expression meaning &quot;straight side.&quot;
According to Google, the Latin plural of &quot;latus rectum&quot; is &quot;latera recta,&quot;
but English allows &quot;latus rectums&quot; or possibly &quot;lati rectums.&quot;
The title of this section is poetry to the eyes and music to the ears of a Latin student
and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic.

The translation of the title is &quot;Latus rectums and other things.&quot; This section describes the calculation of interesting items 
associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines.


When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above.
Then verify that the curve is in fact an ellipse.

From these values everything about the ellipse may be calculated. For example:
{{RoundBoxTop|theme=2}}
[[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.&lt;/math&gt;'''
&lt;/br&gt;
&lt;/br&gt;
Axis : (-0.8)x + (-0.6)y + (9.4) = 0&lt;/br&gt;
Eccentricity = 0.9&lt;/br&gt;
&lt;/br&gt;
Directrix 2 : (0.6)x + (-0.8)y + (2) = 0&lt;/br&gt;
Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0&lt;/br&gt;
Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0&lt;/br&gt;
Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0&lt;/br&gt;
Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{ID2}&lt;/math&gt; = (6.32, 7.24)&lt;/br&gt;
&lt;math&gt;\text{I2}&lt;/math&gt;  = (7.204210526315789473684, 6.061052631578947368421)&lt;/br&gt;
F2  = (8, 5)&lt;/br&gt;
M   = (15.16210526315789473684, -4.54947368421052631579)&lt;/br&gt;
F1  = (22.32421052631578947368, -14.09894736842105263158)&lt;/br&gt;
&lt;math&gt;\text{I1}&lt;/math&gt;  = (23.12, -15.16)&lt;/br&gt;
&lt;math&gt;\text{ID1}&lt;/math&gt; = (24.00421052631578947368, -16.33894736842105263158)&lt;/br&gt;
&lt;/br&gt;
P   = (20.30821052631578947368, -15.61094736842105263158)&lt;/br&gt;
Q   = (10.53708406832736953616, -8.018239580333420216299)&lt;/br&gt;
R   = (5.984, 3.488)&lt;/br&gt;
S   = (10.016, 6.512)&lt;/br&gt;
T   = (19.78712645798841993752, -1.080707788087632415281)&lt;/br&gt;
U   = (24.34021052631578947368, -12.58694736842105263158)&lt;/br&gt;
&lt;/br&gt;
Length of major axis: &lt;math&gt;\text{I1I2}&lt;/math&gt; = 26.52631578947368421052&lt;/br&gt;
Length of minor axis: QT = 11.56255298707631300170&lt;/br&gt;
Length of latus rectum: RS = PU = 5.04
]]
Consider conic section: &lt;math&gt;1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.&lt;/math&gt;

This curve is ellipse with random orientation.
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse

result = calculate_abc_epq(ABCDEF)
(len(result) == 2) or 1/0

# ellipse or hyperbola
(abc1,epq1), (abc2,epq2) = result
a1,b1,c1 = abc1 ; e1,p1,q1 = epq1
a2,b2,c2 = abc2 ; e2,p2,q2 = epq2
(e1 == e2) or 2/0
(1 &gt; e1 &gt; 0) or 3/0

print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) )
A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1)
print ('Equation of ellipse in standard form:')
print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0
Equation of ellipse in standard form:
(-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
# python code
def sum_zero(input) :
    '''
sum = sum_zero(input)
If sum is close to 0 and Tolerance permits, sum is returned as 0.
For example: if input contains (2, -1.999999999999999999999)
this function returns sum of these 2 values as 0.
    '''
    global Tolerance
    sump = sumn = 0
    for v in input :
        if v &gt; 0 : sump += v
        elif v &lt; 0 : sumn -= v
    sum = sump - sumn
    if abs(sum) &lt; Tolerance : return (type(Tolerance))(0)
    min, max = sorted((sumn,sump))
    if abs(sum) &lt;= Tolerance*min : return (type(Tolerance))(0)
    return sum
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
===Major axis===
&lt;syntaxhighlight lang=python&gt;
# axis is perpendicular to directrix.
ax,bx = b1,-a1
# axis contains foci. ax + by + c = 0
cx = reduce_Decimal_number(-(ax*p1 + bx*q1))
axis = ax,bx,cx
print ( '    Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) )
print ( '    Eccentricity = {}'.format(e1) )
print ()
print ( '    Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) )
print ( '    Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) )
F1 = p1,q1 # Focus 1.
print ( '    F1 : ({}, {})'.format(p1,q1) )
F2 = p2,q2 # Focus 2.
print ( '    F2 : ({}, {})'.format(p2,q2) )

#  Direction cosines along axis from F1 towards F2:
dx,dy = a1,b1

# p2 = p1 + dx*distance_F1_F2
# q2 = q1 + dy*distance_F1_F2
if dx : distance_F1_F2 = (p2 - p1)/dx
else :  distance_F1_F2 = (q2 - q1)
if distance_F1_F2 &lt; 0 :
    distance_F1_F2 *= -1
    dx *= -1 ; dy *= -1

# Intercept on directrix1
distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1)
ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1
print ( '    Intercept ID1 : ({}, {})'.format(xID1,yID1) )

# 
# distance_F1_F2
# -------------------- = e
# length_of_major_axis
# 
length_of_major_axis = distance_F1_F2 / e1

# Intercept1 on curve
distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2

xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve
I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ]
print ( '    Intercept I1 : ({}, {})'.format(xI1,yI1) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Axis : (-0.8)x + (-0.6)y + (9.4) = 0
    Eccentricity = 0.9

    Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0
    Directrix 2 : (0.6)x + (-0.8)y + (2) = 0
    F1 : (22.32421052631578947368, -14.09894736842105263158)
    F2 : (8, 5)
    Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158)
    Intercept I1 : (23.12, -15.16)
&lt;/syntaxhighlight&gt;
{{RoundBoxTop|theme=2}}
Techniques similar to above can be used to calculate points 
&lt;math&gt;I2, ID2.&lt;/math&gt;
{{RoundBoxBottom}}

===Latus rectums===
&lt;syntaxhighlight lang=python&gt;
# direction cosines along latus rectum.
dlx,dly = -dy,dx
# 
# distance from U to F1               half_latus_rectum
# ------------------------------ = ----------------------- = e1
# distance from U to directrix 1   distance_from_F1_to_ID1
# 
half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1)
# latus rectum 1
# Focal chord has equation (afc)x + (bfc)y + (cfc) = 0.
afc,bfc = a1,b1
cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1))
print ( '    Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) )
        
P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum
print ( '    Point P : ({}, {})'.format(xP,yP) )
U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum
print ( '    Point U : ({}, {})'.format(xU,yU) )
distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt())
print ('    Length PU =', distance)
print ('    half_latus_rectum =', half_latus_rectum)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0
    Point P : (20.30821052631578947368, -15.61094736842105263158)
    Point U : (24.34021052631578947368, -12.58694736842105263158)
    Length PU = 5.04
    half_latus_rectum = 2.52
&lt;/syntaxhighlight&gt;
{{RoundBoxTop|theme=2}}
Techniques similar to above can be used to calculate points 
&lt;math&gt;R, S.&lt;/math&gt;
{{RoundBoxBottom}}

===Minor axis===
&lt;syntaxhighlight lang=python&gt;
print ()

# Mid point between F1, F2:
M = xM,yM = (p1 + p2)/2, (q1 + q2)/2
print ( '    Mid point M : ({}, {})'.format(xM,yM) )
half_major  = length_of_major_axis / 2
half_distance = distance_F1_F2 / 2

# half_distance**2 + half_minor**2 = half_major**2
half_minor = ( half_major**2 - half_distance**2 ).sqrt()
length_of_minor_axis = half_minor * 2
        
Q = xQ,yQ = xM + dlx*half_minor,  yM + dly*half_minor
T = xT,yT = xM - dlx*half_minor,  yM - dly*half_minor
print ( '    Point Q : ({}, {})'.format(xQ,yQ) )
print ( '    Point T : ({}, {})'.format(xT,yT) )

print ('    length_of_major_axis =', length_of_major_axis)
print ('    length_of_minor_axis =', length_of_minor_axis)
# 
# A basic check.
# length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2)
# 
# length_of_minor_axis**2
# ----------------------- = 1-e**2
# length_of_major_axis**2
# 
# length_of_minor_axis**2
# ----------------------- + (e**2 - 1) = 0
# length_of_major_axis**2
# 
values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1
sum_zero(values) and 3/0
        
aM,bM = a1,b1 # Minor axis is parallel to directrix.
cM = reduce_Decimal_number(-(aM*xM + bM*yM))
print ( '    Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Mid point M : (15.16210526315789473684, -4.54947368421052631579)
    Point Q : (10.53708406832736953616, -8.018239580333420216299)
    Point T : (19.78712645798841993752, -1.080707788087632415281)
    length_of_major_axis = 26.52631578947368421052
    length_of_minor_axis = 11.56255298707631300170
    Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

===Checking===
{{RoundBoxTop|theme=2}}
All interesting points have been calculated without using equations of any of the relevant lines.

However, equations of relevant lines are very useful for testing, for example:


* Check that points &lt;math&gt;ID2, I2, F2, M, F1, I1, ID1&lt;/math&gt; are on axis.

* Check that points &lt;math&gt;R, F2, S&lt;/math&gt; are on latus rectum through &lt;math&gt;F2.&lt;/math&gt;

* Check that points &lt;math&gt;Q, M, T&lt;/math&gt; are on minor axis through &lt;math&gt;M.&lt;/math&gt;

* Check that points &lt;math&gt;P, F1, U&lt;/math&gt; are on latus rectum through &lt;math&gt;F1.&lt;/math&gt;


Test below checks that 8 points &lt;math&gt;I1, I2, P, Q, R, S, T, U&lt;/math&gt; are on ellipse and satisfy eccentricity &lt;math&gt;e = 0.9.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
t1 = (
    ('I1'), ('I2'),
    ('P'), ('Q'), ('R'),
    ('S'), ('T'), ('U'),
)

for name in t1 :
    value = eval(name)
    x,y = [ reduce_Decimal_number(v) for v in value ]
    print ('{} : ({}, {})'.format((name+' ')[:2], x,y))
    values = A*x**2, B*y**2, C*x*y, D*x, E*y, F
    sum_zero(values) and 3/0

    # Relative to Directrix 1 and Focus 1:
    distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt()
    distance_to_directrix1 = a1*x + b1*y + c1
    e1 = distance_to_F1 / distance_to_directrix1
    print ('    e1 =',e1) # Raw value is printed.

    # Relative to Directrix 2 and Focus 2:
    distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt()
    distance_to_directrix2 = a2*x + b2*y + c2
    e2 = distance_to_F2 / distance_to_directrix2
    e2 = reduce_Decimal_number(e2)
    print ('    e2 =',e2) # Clean value is printed.
&lt;/syntaxhighlight&gt;
{{RoundBoxTop|theme=2}}
Note the differences between &quot;raw&quot; values of &lt;math&gt;e_1&lt;/math&gt; and &quot;clean&quot; values of &lt;math&gt;e_2.&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
I1 : (23.12, -15.16)
    e1 = -0.9000000000000000000034 
    e2 = 0.9
I2 : (7.204210526315789473684, 6.061052631578947368421)
    e1 = -0.9 
    e2 = 0.9
P  : (20.30821052631578947368, -15.61094736842105263158) 
    e1 = -0.9 
    e2 = 0.9
Q  : (10.53708406832736953616, -8.018239580333420216299)
    e1 = -0.9000000000000000000002
    e2 = 0.9
R  : (5.984, 3.488)
    e1 = -0.9000000000000000000003
    e2 = 0.9
S  : (10.016, 6.512) 
    e1 = -0.9000000000000000000003
    e2 = 0.9   
T  : (19.78712645798841993752, -1.080707788087632415281)
    e1 = -0.8999999999999999999996
    e2 = 0.9
U  : (24.34021052631578947368, -12.58694736842105263158)
    e1 = -0.9
    e2 = 0.9
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

==Traditional definition of ellipse==
{{RoundBoxTop|theme=2}}
[[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.'''
&lt;/br&gt;
Sum of distance &lt;math&gt;F_1G&lt;/math&gt; and distance &lt;math&gt;F_2G&lt;/math&gt; is constant.
]]
Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant.


In the diagram the two fixed points are the foci, Focus 1 or &lt;math&gt;F_1&lt;/math&gt; and Focus 2 or &lt;math&gt;F_2.&lt;/math&gt;

Distance between &lt;math&gt;F_1&lt;/math&gt; and &lt;math&gt;F_2&lt;/math&gt;, distance &lt;math&gt;F_1F_2&lt;/math&gt;, must be non-zero.

Point &lt;math&gt;G&lt;/math&gt; on perimeter of ellipse moves so that sum of distance &lt;math&gt;F_1G&lt;/math&gt;
and distance &lt;math&gt;F_2G&lt;/math&gt; is constant.


Points &lt;math&gt;T_1&lt;/math&gt; and &lt;math&gt;T_2&lt;/math&gt; are on axis of ellipse and the same rule applies to these points.

distance &lt;math&gt;F_1T_1&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt; is constant.


distance &lt;math&gt;F_1T_1&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt;

&lt;math&gt;=&lt;/math&gt; distance &lt;math&gt;F_1G&lt;/math&gt; + distance &lt;math&gt;F_2G&lt;/math&gt;

&lt;math&gt;=&lt;/math&gt; distance &lt;math&gt;F_2T_2&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt;

&lt;math&gt;= \text{length of major axis.}&lt;/math&gt;

Therefore the constant is &lt;math&gt;\text{length of major axis}&lt;/math&gt; which must be greater than distance &lt;math&gt;F_1F_2.&lt;/math&gt;


From information given, calculate eccentricity &lt;math&gt;e&lt;/math&gt; and equation of one directrix.
Choose directrix 1 &lt;math&gt;dx1&lt;/math&gt; associated with focus F1.


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
==Ellipse at origin==
{{RoundBoxTop|theme=2}}
[[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.'''
&lt;/br&gt;
Sum of distance &lt;math&gt;F_1P&lt;/math&gt; and distance &lt;math&gt;F_2P&lt;/math&gt; is constant.
]]
Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant.


By definition distance &lt;math&gt;F_2P&lt;/math&gt; + distance &lt;math&gt;F_1P&lt;/math&gt; is constant.

&lt;math&gt;\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)&lt;/math&gt;

Expand &lt;math&gt;(1)&lt;/math&gt; and result is &lt;math&gt;Ax^2 + By^2 + F = 0\ \dots\ (2)&lt;/math&gt; where:

&lt;math&gt;A = 4k^2 - 16p^2&lt;/math&gt;

&lt;math&gt;B = 4k^2&lt;/math&gt;

&lt;math&gt;F = 4k^2p^2 - k^4&lt;/math&gt;


When &lt;math&gt;y = 0,&lt;/math&gt; point &lt;math&gt;B,\ Ax^2 = -F&lt;/math&gt;

&lt;math&gt;x^2 = \frac{-F}{A}&lt;/math&gt;
&lt;math&gt;= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}&lt;/math&gt;
&lt;math&gt;=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.&lt;/math&gt;

Therefore: &lt;math&gt;x = \frac{k}{2} = a&lt;/math&gt;

&lt;math&gt;k = \text{length of major axis.}&lt;/math&gt;

By definition, distance &lt;math&gt;F_2A&lt;/math&gt; + distance &lt;math&gt;F_1A = k.&lt;/math&gt;

Therefore distance &lt;math&gt;F_1A = a.&lt;/math&gt;


Intercept form of ellipse at origin:

&lt;math&gt;(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2&lt;/math&gt;

&lt;math&gt;\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1&lt;/math&gt;

&lt;math&gt;\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1&lt;/math&gt;

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

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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==Second definition of ellipse==
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[[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt; where &lt;math&gt;a,b = 20,12&lt;/math&gt;.'''
&lt;/br&gt;
At point &lt;math&gt;B,\ \frac{u}{v} = e.&lt;/math&gt;
&lt;/br&gt;
At point &lt;math&gt;A,\ \frac{a}{t} = e.&lt;/math&gt;
]]
Ellipse is path of point that moves so that ratio of distance to
fixed point and distance to fixed line is constant.


Let &lt;math&gt;\frac{p}{a} = e&lt;/math&gt; where:

* &lt;math&gt;p&lt;/math&gt; is non-zero,

* &lt;math&gt;a &gt; p,&lt;/math&gt;

* &lt;math&gt;a = p + u.&lt;/math&gt;

Therefore, &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;

Let directrix have equation &lt;math&gt;x = t&lt;/math&gt; where &lt;math&gt;\frac{a}{t} = e.&lt;/math&gt;

At point &lt;math&gt;B:&lt;/math&gt;

&lt;math&gt;\frac{p}{p+u} = \frac{p+u}{p+u+v} = e&lt;/math&gt;

&lt;math&gt;(p+u)^2 = p(p+u+v)&lt;/math&gt;

&lt;math&gt;pp + pu + pu + uu = pp + pu + pv&lt;/math&gt;

&lt;math&gt;pu + uu = pv&lt;/math&gt;

&lt;math&gt;u(p + u) = pv&lt;/math&gt;

&lt;math&gt;\frac{u}{v} = \frac{p}{p+u} = e&lt;/math&gt;

&lt;math&gt;\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)&lt;/math&gt;

Statement &lt;math&gt;(3)&lt;/math&gt; is true at point &lt;math&gt;A&lt;/math&gt; also.


Section under &quot;Proof&quot; below proves that statement (3) is true for any point &lt;math&gt;P&lt;/math&gt; on ellipse.
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===Proof===
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[[File:0902ellipse00.png|thumb|400px|'''Proving that &lt;math&gt;\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e&lt;/math&gt;.'''
&lt;/br&gt;
Graph is part of curve &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.&lt;/math&gt;
&lt;/br&gt;
distance to Directrix1 &lt;math&gt;= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.&lt;/math&gt;
&lt;/br&gt;
base = &lt;math&gt;x - p = x - ae&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{(distance to Focus1)}^2 = \text{base}^2 + y^2&lt;/math&gt;
]]
As expressed above in statement &lt;math&gt;3,&lt;/math&gt; second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to
fixed point and distance to fixed line is constant.


This section proves that this definition is true for any point &lt;math&gt;P&lt;/math&gt; on the ellipse.


At point &lt;math&gt;P:&lt;/math&gt;

&lt;math&gt;(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0&lt;/math&gt;

&lt;math&gt;y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}&lt;/math&gt;

&lt;math&gt;= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}&lt;/math&gt;

&lt;math&gt;= e^2x^2 - x^2 + a^2 - a^2e^2&lt;/math&gt;

base &lt;math&gt;= x-p = x-ae&lt;/math&gt;

&lt;math&gt;(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2&lt;/math&gt;
&lt;math&gt;= a^2 - 2aex + e^2x^2&lt;/math&gt;
&lt;math&gt;= (a-ex)^2&lt;/math&gt;

&lt;math&gt;\text{distance to Focus1} = \text{distance}\ F_1P = a - ex&lt;/math&gt;

&lt;math&gt;\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}&lt;/math&gt;

&lt;math&gt;\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}&lt;/math&gt;
&lt;math&gt;= (a - ex)\frac{e}{(a-ex)}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;


Similar calculations can be used to prove the case for Focus2 &lt;math&gt;(-p, 0)&lt;/math&gt; and Directrix2 &lt;math&gt;(x = -t)&lt;/math&gt;
in which case:

&lt;math&gt;\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}&lt;/math&gt;
&lt;math&gt;= (a + ex)\frac{e}{(a + ex)}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;

Therefore: &lt;math&gt;\frac{\text{distance to focus}}{\text{distance to directrix}} = e&lt;/math&gt; where &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;


Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity &lt;math&gt;e.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

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&lt;/syntaxhighlight&gt;
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&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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==Heading==
===Properties of ellipse===
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[[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.'''
&lt;/br&gt;
&lt;/br&gt;
Traditional definition of ellipse:
&lt;/br&gt;
&lt;math&gt;\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Second definition of ellipse:
&lt;/br&gt;
&lt;math&gt;\frac{\text{distance } AF_1}  {\text{distance } AG } = \text{eccentricity } e.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Triangle &lt;math&gt;A F_1  G&lt;/math&gt; is right triangle.
&lt;/br&gt;
&lt;math&gt;e = \cos \angle  O F_1  A  = \cos \angle   F_1 A G&lt;/math&gt;
]]
Ellipse in diagram has:


* Two foci: &lt;math&gt;F_1\ (p,0),\ F_2\ (-p,0).&lt;/math&gt;

* Length of major axis &lt;math&gt;= \text{distance } I_2 I_1 = 2a&lt;/math&gt;

* Length of minor axis &lt;math&gt;= \text{distance } A B = 2b&lt;/math&gt;

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

* Length of latus rectum &lt;math&gt;= \text{distance } P Q&lt;/math&gt;

* Distance between directrices &lt;math&gt;= \text{distance } D_2 D_1 = 2t&lt;/math&gt;


Properties of ellipse:

* &lt;math&gt;\frac{\text{length of major axis}}  {\text{distance between directrices}} = e&lt;/math&gt;


* &lt;math&gt;\frac{\text{distance between foci}}  {\text{length of major axis}} = e&lt;/math&gt;


* &lt;math&gt;\frac{\text{distance between foci}}  {\text{distance between directrices}}= e^2&lt;/math&gt;


* &lt;math&gt;(\frac{\text{length of minor axis}}  {\text{length of major axis}})^2 + e^2 = 1&lt;/math&gt;


* &lt;math&gt;\frac{\text{length of latus rectum}}  {\text{length of major axis}} + e^2 = 1&lt;/math&gt;


* line &lt;math&gt;PD_1&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;
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&lt;/syntaxhighlight&gt;
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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====Major axis====
From traditional definition of ellipse:

Distance &lt;math&gt;AF_2\ +&lt;/math&gt; distance &lt;math&gt;AF_1&lt;/math&gt;

= distance &lt;math&gt;I_1F_1\ +&lt;/math&gt; distance &lt;math&gt;I_1F_2&lt;/math&gt;

= distance &lt;math&gt;I_2F_2\ +&lt;/math&gt; distance &lt;math&gt;I_2F_1&lt;/math&gt;

= &lt;math&gt;k.&lt;/math&gt;


Therefore: 

Length of major axis = distance &lt;math&gt;I_2I_1 = 2a = k.&lt;/math&gt;

Distance &lt;math&gt;AF_1 = \frac{k}{2} = a.&lt;/math&gt;


From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e&lt;/math&gt;
&lt;math&gt;= \frac{\text{distance }OI_1}{\text{distance }OD_1}.&lt;/math&gt;

&lt;math&gt;\frac{\text{length of major axis}}{\text{distance between directrices}} = e.&lt;/math&gt;
====Foci====
From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.&lt;/math&gt;

&lt;math&gt;a - p = te - ae&lt;/math&gt;

&lt;math&gt;a - p = a - ae&lt;/math&gt;

Therefore: &lt;math&gt;p = ae&lt;/math&gt; or &lt;math&gt;\frac{p}{a} = e.&lt;/math&gt;

&lt;math&gt;\frac{\text{distance between foci}}{\text{length of major axis}} = e.&lt;/math&gt;

&lt;math&gt;\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.&lt;/math&gt;
====Minor axis====
Triangle &lt;math&gt;AOF_1&lt;/math&gt; is right triangle.

&lt;math&gt;\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1&lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    (\frac{p}{a})^2  &lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    (\frac{ae}{a})^2  &lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    e^2  = 1&lt;/math&gt;


&lt;math&gt;( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1&lt;/math&gt;


Triangles &lt;math&gt;AOF_1,\ AF_1G&lt;/math&gt; are similar.

Triangle &lt;math&gt;AF_1G&lt;/math&gt; is right triangle.


&lt;math&gt;e = \cos \angle OF_1A = \cos \angle F_1AG.&lt;/math&gt;
====Latus rectum====
From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e&lt;/math&gt;

&lt;math&gt;\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).&lt;/math&gt;

&lt;math&gt;\frac{\text{distance }PF_1} {a} = 1 - e^2.&lt;/math&gt;

&lt;math&gt; \frac{\text{length of latus rectum}} {\text{length of major axis}}  + e^2 = 1&lt;/math&gt;

====Slope of curve====
Curve has equation: &lt;math&gt;b^2x^2 + a^2y^2 = a^2b^2&lt;/math&gt;

Derivative of both sides:

&lt;math&gt;b^22x + a^22yy' = 0&lt;/math&gt;

&lt;math&gt;y' = \frac{-xb^2}{ya^2}&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;math&gt;= \frac{-x(1-e^2)}{y}&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

At point &lt;math&gt;P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}&lt;/math&gt;
&lt;math&gt;= \frac{ae}{a} = e.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

Slope of line &lt;math&gt;PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

&lt;math&gt;m_1 = m_2.&lt;/math&gt; Therefore line &lt;math&gt;PD_1&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;

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

===Intercept form of equation===


&lt;math&gt;\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.'''
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
Equation of ellipse has format &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt; where:
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{Length of minor axis} = 2b = \text{distance}\ BA = 24&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
]]
In diagram:

Intercept &lt;math&gt;I_1&lt;/math&gt; has coordinates &lt;math&gt;(a,0).&lt;/math&gt;

Intercept &lt;math&gt;I_2&lt;/math&gt; has coordinates &lt;math&gt;(-a,0).&lt;/math&gt;

Intercept &lt;math&gt;A&lt;/math&gt; has coordinates &lt;math&gt;(0,b).&lt;/math&gt;

Intercept &lt;math&gt;B&lt;/math&gt; has coordinates &lt;math&gt;(0,-b).&lt;/math&gt;

Focus &lt;math&gt;F_1&lt;/math&gt; has coordinates &lt;math&gt;(f,0)&lt;/math&gt; where &lt;math&gt;f = ea.&lt;/math&gt;

Focus &lt;math&gt;F_2&lt;/math&gt; has coordinates &lt;math&gt;(-f,0).&lt;/math&gt;

Curve has equation &lt;math&gt;\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,&lt;/math&gt; called intercept form of equation of ellipse because intercepts are apparent as the 
fractional value of each coefficient.


Standard form of this equation is: &lt;math&gt;(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.&lt;/math&gt; While the standard form is
valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and
intercepts are neatly contained within the equation.


Slope of curve:

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

Derivative of both sides:

&lt;math&gt;b^22x + a^22yy' = 0&lt;/math&gt;

&lt;math&gt;y' = \frac{-xb^2}{ya^2}&lt;/math&gt;
&lt;math&gt;= \frac{-x(1-e^2)}{y}&lt;/math&gt;


At point &lt;math&gt;P&lt;/math&gt; on latus rectum &lt;math&gt;PQ:&lt;/math&gt;

&lt;math&gt;m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e&lt;/math&gt;


Slope of line &lt;math&gt;PD = m_2 = \frac{PF_1}{F_1D} = e&lt;/math&gt;

&lt;math&gt;m_1 = m_2.&lt;/math&gt; Line &lt;math&gt;PD&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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&lt;math&gt;&lt;/math&gt;
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&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
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===Example===
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[[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.'''
&lt;/br&gt;
M is mid-point between F1 and F2.
&lt;/br&gt;
Point R is on minor axis.
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{F1:}\ (1, -7)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{F2:}\ (-1.24, 0.68)&lt;/math&gt;
&lt;/br&gt;
length_of_major_axis = 10
&lt;/br&gt;
&lt;math&gt;\text{M:}\ (-0.12, -3.16)&lt;/math&gt;
&lt;/br&gt;
length_of_minor_axis = 6
&lt;/br&gt;
&lt;math&gt;\text{R:}\ (2.76, -2.32)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e = 0.8&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{D1:}\ (1.63, -9.16)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{epq}\ =\ (0.8,\ 1,\ -7)&lt;/math&gt;
]]
Given:
&lt;syntaxhighlight lang=python&gt;
# python code

F1 = 1, -7          # Focus 1
F2 = -1.24, 0.68    # Focus 2
length_of_major_axis = 10
&lt;/syntaxhighlight&gt;
Calculate equation of ellipse.
&lt;syntaxhighlight lang=python&gt;

F1 = p1,q1 = [ dD(str(v)) for v in F1 ]    # Focus 1
F2 = p2,q2 = [ dD(str(v)) for v in F2 ]    # Focus 2

length_of_major_axis = dD(length_of_major_axis)
half_major_axis = length_of_major_axis / 2

# Direction cosines from F1 to F2
dx = p2-p1 ; dy = q2-q1
divider = (dx**2 + dy**2).sqrt()
dx,dy = [ (v/divider) for v in (dx,dy) ]

# F2
# p2 = p1 + dx*distance_F1_F2
# q2 = q1 + dy*distance_F1_F2
distance_F1_F2 = (q2-q1)/dy
half_distance_F1_F2 = distance_F1_F2 / 2

# The mid-point
M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2

# Eccentricity:
e = distance_F1_F2 / length_of_major_axis

# distance from point R to F1                half_major_axis
# ------------------------------------ = e = -----------------------------------------
# distance from point R to Directrix 1       distance from point M to Directrix 1

distance_from_point_M_to_dx1 = half_major_axis / e

# Intersection of axis and directrix 1
D1 = xM-dx*distance_from_point_M_to_dx1,  yM-dy*distance_from_point_M_to_dx1
D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ]

# Equation of Directrix 1
# dx1 = adx1,bdx1,cdx1
adx1,bdx1 = dx, dy # Perpendicular to axis.
# adx1*x + bdx1*y + cdx1 = 0
# Directrix 1 contains point D1
cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) )

abc = adx1,bdx1,cdx1
epq = e,p1,q1
ABCDEF = ABCDEF_from_abc_epq  (abc,epq, 1)
&lt;/syntaxhighlight&gt;
Equation of ellipse in standard form:

&lt;math&gt;(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0&lt;/math&gt;


For more insight into method of calculation and proof:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    print ('F1: ({}, {})'.format(p1,q1))
    print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1))

    print ('F2: ({}, {})'.format(p2,q2))
    print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2))

    print ('length_of_major_axis =', length_of_major_axis)
    
    print ('M: ({}, {})'.format(xM,yM))
    print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM))

    # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2
    half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt()
    length_of_minor_axis = half_minor_axis * 2
    s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1))

    # Direction cosines on major axis:
    print ('dx,dy =', dx,dy)
    
    # Direction cosines on minor axis:
    dnx,dny = dy,-dx
    print ('dnx,dny =', dnx,dny)

    # One point on minor axis:
    R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis
    print ('R: ({}, {})'.format(xR,yR))
    print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR))
    # Verify that point R is on ellipse:
    sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0

    s1 = 'e' ; print (s1, '=', eval(s1))

    print ('D1: ({}, {})'.format(xD1,yD1))
    print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1))

    print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1))

    print()
# For proof, reverse the process:    
    (abc1,epq1), (abc2,epq2) = calculate_abc_epq  (ABCDEF)
    a1,b1,c1 = abc1 ; e1,p1,q1 = epq1
    print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1))
    print ('Eccentricity e1: {}'.format(e1))
    print ('F1: ({}, {})'.format(p1,q1))

    print()
    a2,b2,c2 = abc2 ; e2,p2,q2 = epq2
    print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2))
    print ('Eccentricity e2: {}'.format(e2))
    print ('F2: ({}, {})'.format(p2,q2))

    print ('\nEquation of ellipse with integer coefficients:')
    A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ]
    str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'
    print (str1.format(A,B,C,D,E,F))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
F1: (1, -7)
F1: (x - (1))^2 + (y - (-7))^2 = 1
F2: (-1.24, 0.68)
F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1
length_of_major_axis = 10
M: (-0.12, -3.16)
M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1
length_of_minor_axis = 6
dx,dy = -0.28 0.96
dnx,dny = 0.96 0.28
R: (2.76, -2.32)
R: (x - (2.76))^2 + (y - (-2.32))^2 = 1
e = 0.8
D1: (1.63, -9.16)
D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1
Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0

Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0
Eccentricity e1: 0.8
F1: (1, -7)

Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 
Eccentricity e2: 0.8
F2: (-1.24, 0.68) 

Equation of ellipse with integer coefficients: 
&lt;/syntaxhighlight&gt;
&lt;math&gt;(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

=allEqual=

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;
{| class=&quot;wikitable&quot;
|-
! &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2 - N&lt;/math&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;6&lt;/code&gt; || &lt;code&gt;-221&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;7&lt;/code&gt; || &lt;code&gt;-208&lt;/code&gt;
|-
|-
| &lt;code&gt;10&lt;/code&gt; || &lt;code&gt;-157&lt;/code&gt;
|-
| &lt;code&gt;11&lt;/code&gt; || &lt;code&gt;-136&lt;/code&gt;
|-
| &lt;code&gt;12&lt;/code&gt; || &lt;code&gt;-113&lt;/code&gt;
|-
| &lt;code&gt;13&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-88&lt;/code&gt;
|-
| &lt;code&gt;26&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;419&lt;/code&gt;
|}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;</text>
      <sha1>s49j76f66cdfrh8zjsxckym2e2dqj3q</sha1>
    </revision>
    <revision>
      <id>2815177</id>
      <parentid>2815176</parentid>
      <timestamp>2026-06-11T08:42:09Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <minor />
      <comment>/* Complex Cube Root */</comment>
      <origin>2815177</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="56863" sha1="7v48256re8rlvwwl5muohsgn80au6b0" xml:space="preserve">

==Table==

{| class=&quot;wikitable&quot;
|-
| From &lt;math&gt;(3):&lt;/math&gt; || &lt;math&gt;p^3&lt;/math&gt;|| || &lt;math&gt;+Ap&lt;/math&gt;  || &lt;math&gt;+B&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (5)&lt;/math&gt;
|-
| From &lt;math&gt;(4):&lt;/math&gt; || || &lt;math&gt;Cp^2&lt;/math&gt;|| &lt;math&gt;&lt;/math&gt;  || &lt;math&gt;+D&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (6)&lt;/math&gt;
|-
| &lt;math&gt;(5)*D&lt;/math&gt; || &lt;math&gt;Dp^3&lt;/math&gt;|| || &lt;math&gt;+DAp&lt;/math&gt;  || &lt;math&gt;+DB&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (7)&lt;/math&gt;
|-
| &lt;math&gt;(6)*B&lt;/math&gt; || || &lt;math&gt;BCp^2&lt;/math&gt;|| &lt;math&gt;&lt;/math&gt;  || &lt;math&gt;+BD&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (8)&lt;/math&gt;
|-
| &lt;math&gt;(7)-(8)&lt;/math&gt; || &lt;math&gt;Dp^3&lt;/math&gt;||&lt;math&gt;-BCp^2&lt;/math&gt; || &lt;math&gt;+DAp&lt;/math&gt;  || &lt;math&gt;&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (9)&lt;/math&gt;
|-
| Simplify &lt;math&gt;(9)&lt;/math&gt; || &lt;math&gt;&lt;/math&gt;||&lt;math&gt;Dp^2&lt;/math&gt; || &lt;math&gt;-BCp&lt;/math&gt;  || &lt;math&gt;+DA&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (10)&lt;/math&gt;
|-
| &lt;math&gt;(6)*A&lt;/math&gt; || || &lt;math&gt;ACp^2&lt;/math&gt;|| &lt;math&gt;&lt;/math&gt;  || &lt;math&gt;+AD&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (11)&lt;/math&gt;
|-
| &lt;math&gt;(10)-(11)&lt;/math&gt; || &lt;math&gt;&lt;/math&gt;||&lt;math&gt;Dp^2-ACp^2&lt;/math&gt; || &lt;math&gt;-BCp&lt;/math&gt;  || &lt;math&gt;&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (12)&lt;/math&gt;
|-
| Simplify &lt;math&gt;(12)&lt;/math&gt; || &lt;math&gt;&lt;/math&gt;|| || &lt;math&gt;Dp-ACp&lt;/math&gt; || &lt;math&gt;-BC&lt;/math&gt;  || &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (13)&lt;/math&gt;
|}

==Implementation==

&lt;math&gt;(1)&lt;/math&gt; squared: &lt;math&gt;a^2 = p^6 - 6p^4q^2\ + 9p^2q^4&lt;/math&gt;

&lt;math&gt;p^2 p^2 p^2 - (6) p^2 p^2 q^2\ + (9)p^2q^4 - a^2\dots\ (1a)&lt;/math&gt;

From &lt;math&gt;(2):\ 3p^2q = b + q^3\ \dots\ (2a)&lt;/math&gt;


&lt;math&gt;(1a) * 27q^3:&lt;/math&gt;

&lt;math&gt;27q^3a^2 = 27q^3p^6 - 27q^3(6)p^4q^2\ + 27q^3(9)p^2q^4&lt;/math&gt;

&lt;math&gt;27q^3a^2 = 27(p^2q)^3 - 27(6)p^4q^5\ + 27(9)p^2q^7&lt;/math&gt;

&lt;math&gt;27q^3a^2 = (3p^2q)^3 - 27(6)p^4q^2q^3\ + 27(9)p^2qq^6&lt;/math&gt;

&lt;math&gt;27q^3a^2 = (3p^2q)^3 - 3(6)(9p^4q^2)q^3\ + 27(3)(3p^2q)q^6&lt;/math&gt;

&lt;math&gt;27q^3a^2 = (3p^2q)^3 - 3(6)((3p^2q)^2)q^3\ + 27(3)(3p^2q)q^6&lt;/math&gt;

Let &lt;math&gt;Q = q^3:&lt;/math&gt;

&lt;math&gt;27Qa^2 = (3p^2q)^3 - 3(6)((3p^2q)^2)Q + 27(3)(3p^2q)Q^2\ \dots\ (1b)&lt;/math&gt;


For &lt;math&gt;(3p^2q)&lt;/math&gt; in &lt;math&gt;(1b)&lt;/math&gt; substitute &lt;math&gt; ( b + Q ) : &lt;/math&gt;

&lt;math&gt;27Qa^2 = (b+Q)^3 - 3(6)((b+Q)^2)Q + 27(3)(b+Q)Q^2\ \dots\ (1c)&lt;/math&gt;

==Complex Cube Root==

See [https://en.wikiversity.org/wiki/Cubic_function#Vieta's_substitution Vieta's Substitution.]

The depressed cubic function is : &lt;math&gt;f(t) = t^3 + At + B = 0.&lt;/math&gt;

Let &lt;math&gt;A = -3C&lt;/math&gt; and let &lt;math&gt;t = w + \frac{C}{w} = \frac{w^2 + C}{w}.&lt;/math&gt;

Substitute for &lt;math&gt;A, t&lt;/math&gt; in the depressed function and result is:

&lt;math&gt;f(w) = w^6 + Bw^3 + C^3&lt;/math&gt; or

&lt;math&gt;f(W) = W^2 + BW + C^3&lt;/math&gt; where &lt;math&gt;W = w^3&lt;/math&gt; and &lt;math&gt;w = \sqrt[3]{W}&lt;/math&gt;.

From the quadratic formula: &lt;math&gt;W = \frac{-B \pm \sqrt{B^2 - 4C^3}}{2} = \frac{-B \pm \sqrt{4C^3 - B^2}\sqrt{-1}   }{2}&lt;/math&gt;

However, &lt;math&gt;W = a + bi.&lt;/math&gt;

Therefore: &lt;math&gt;a = \frac{-B}{2}&lt;/math&gt; and &lt;math&gt;b = \frac{ \sqrt{4C^3 - B^2} }{2}.&lt;/math&gt;

&lt;math&gt;B = -2a&lt;/math&gt; and

&lt;math&gt;2b = \sqrt{4C^3 - B^2} &lt;/math&gt;

&lt;math&gt;4b^2 = 4C^3 - B^2 &lt;/math&gt;

&lt;math&gt;4C^3 = 4b^2 + B^2 &lt;/math&gt;

&lt;math&gt;C^3 = b^2 + \frac{B^2}{4} &lt;/math&gt;

&lt;math&gt;C = \sqrt[3]{C^3} &lt;/math&gt; and

&lt;math&gt;A = -3C. &lt;/math&gt;

&lt;math&gt;A,B &lt;/math&gt; of &lt;math&gt;f(t)&lt;/math&gt; have been defined with &lt;math&gt;C&lt;/math&gt; positive and &lt;math&gt;A&lt;/math&gt; negative.

Because &lt;math&gt;W&lt;/math&gt; has three complex roots, &lt;math&gt;f(t)&lt;/math&gt; must have three real roots.

Calculate one of the roots of &lt;math&gt;f(t).&lt;/math&gt;

&lt;math&gt;t = \frac{w^2 + C}{w}.&lt;/math&gt;

Therefore &lt;math&gt;wt = w^2 + C&lt;/math&gt; and

&lt;math&gt;w^2 - tw + C = 0.&lt;/math&gt;

From the quadratic formula : 

&lt;math&gt;w = \frac{t \pm \sqrt{t^2 - 4C}}{2} = \frac{t \pm \sqrt{4C - t^2}\sqrt{-1}   }{2}&lt;/math&gt;

&lt;math&gt;w = p + qi&lt;/math&gt;

Therefore: &lt;math&gt;p = \frac{t}{2}&lt;/math&gt; and &lt;math&gt;Q = \frac{4C - t^2  }{4} = C - \frac{t^2}{4}&lt;/math&gt; where &lt;math&gt;Q = q^2.&lt;/math&gt;

&lt;math&gt;q = \sqrt{Q}&lt;/math&gt; but sign of this calculation of &lt;math&gt;q&lt;/math&gt; is ambiguous.

Let &lt;math&gt;P = p^2&lt;/math&gt;

&lt;math&gt;q = \frac{b}{(3P - Q)}&lt;/math&gt;

and &lt;math&gt;(p + qi)^3 = a + bi.&lt;/math&gt;
===An Example===

Calculate cube roots of complex number &lt;math&gt;W = -39582 + 3799i.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:1215depressed cubic01.png|thumb|400px|'''Graph of &lt;math&gt;f(t)&lt;/math&gt; shown as graph of &lt;math&gt;f(x)&lt;/math&gt; and showing three values of &lt;math&gt;t: t_1, t_2, t_3&lt;/math&gt;.'''
&lt;/br&gt;
&lt;math&gt;Y&lt;/math&gt; axis compressed for clarity.
&lt;/br&gt;
&lt;math&gt;A,B = -3495.0, 79164.0&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;t_1 = -68.229473419497441512295943903670298\dots&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;t_2 = 32.229473419497441512295943903670298\dots&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\ t_3 = 36&lt;/math&gt;
]]

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

import decimal
decimal.getcontext().prec = 22
ab = -39582,3799
a,b = [ dD(v) for v in ab ]
B = -2*a
C = (b**2 + B*B/4) ** (dD(1)/3)
A = -3*C

a,b,A,B,C = [ dD(str(float(v))) for v in (a,b,A,B,C) ]
print ( 'a,b = {}, {}'.format(a,b) )
print ( 'A,B,C = {}, {}, {}'.format(A,B,C) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
a,b = -39582.0, 3799.0
A,B,C = -3495.0, 79164.0, 1165.0
&lt;/syntaxhighlight&gt;
Calculate roots of cubic function:

&lt;math&gt;y = f (t) &lt;/math&gt;&lt;math&gt; = t^3 &lt;/math&gt;&lt;math&gt; - 3495 t &lt;/math&gt;&lt;math&gt; + 79164 .&lt;/math&gt;

Three roots are: &lt;math&gt;t_1,\ t_2,\ t_3 = -68.22947341949744\dots,\ 32.22947341949744\dots,\ 36.0&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
# python code:
t1, t2, t3 = ('-68.229473419497441512295943903670298',
               '32.229473419497441512295943903670298', 36 )
values_of_t = [ dD(v) for v in (t1,t2,t3) ]

for t in values_of_t :
    p = t/2 ; P = p**2
    Q = C - t*t/4 ; q = b/(3*P - Q)
    # Check results:
    print ()
    sx = 't' ; print (sx,'=', eval(sx))
    print ( 'p,q = {}, {}'.format(p,q) )
    ab = [ p*P - 3*p*Q, 3*P*q - q*Q ]
    a_,b_ = [ float(v) for v in ab ]
    sx = 'a_,b_' ; print (sx,'=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
t = -68.229473419497441512295943903670298
p,q = -34.11473670974872075615, 1.088457268119895641747
a_,b_ = (-39582.0, 3799.0)
    
t = 32.229473419497441512295943903670298
p,q = 16.11473670974872075615, -30.08845726811989564176
a_,b_ = (-39582.0, 3799.0)
    
t = 36
p,q = 18, 29
a_,b_ = (-39582.0, 3799.0)
&lt;/syntaxhighlight&gt;
Three cube roots of &lt;math&gt;W = -39582 + 3799i&lt;/math&gt; are:

&lt;math&gt;w_1 = -34.11473670974872075615 + 1.088457268119895641747i&lt;/math&gt;

&lt;math&gt;w_2 = 16.11473670974872075615 - 30.08845726811989564176i&lt;/math&gt;

&lt;math&gt;w_3 = 18 + 29i&lt;/math&gt;

==Making the Decimal object==

The following function verifies that we are working with Decimal objects. 

&lt;syntaxhighlight lang=python&gt;
import sys
import decimal

getcontext = decimal.getcontext
dD = D = Decimal = decimal.Decimal
DT = decimal.DecimalTuple
dgt = decimal.getcontext()

dgt.prec = 50


def print_error (error, x=None, thisName=None) :
    &quot;&quot;&quot;
Prints error derived from sys.exc_info().
    &quot;&quot;&quot;
    list1 = error.split(',')
    v1 = ','.join(list1[1:-1])
    if thisName : print (thisName)
    if x != None : print ('  Input =', (str(x))[:60])
    print (' ', list1[0])
    print (' ', v1)
    print (' ', list1[-1])


def makeDecimal (x, flag = 0) :
    '''
output = makeDecimal (x [, flag]) 
x is a single object convertible to Decimal object.
returns Decimal object or
returns None on error.
'''
    thisName = 'makeDecimal (x) :'
    if flag :
        print ()
        print (thisName)
        print ('  Input x =', type(x), (str(x))[:60])
        
    if isinstance(x, dD) : return x
    if isinstance(x, int) : return dD(x)
    if isinstance(x, float) : return dD(str(x))
    if isinstance(x, str) :
        try :
            error = ''
            output = dD(x)
        except : error = str(sys.exc_info())
        if error :
            if flag : print_error (error) 
            return None
        return output
        
    if isinstance(x, (list,tuple,DT)) :
        try :
            error = ''
            v1,v2,v3 = x
            if isinstance(v2, (tuple,list)) : output = dD(x)
            else : output = dD( ( v1, list(v2), v3 ) )
        except : error = str(sys.exc_info())
        if error :
            if flag : print_error (error) 
            return None
        return output

    if flag :
        print ('  Input not recognized.')
    return None
&lt;/syntaxhighlight&gt;
===checkComplex(x)===

This function verifies that the object is a valid complex tuple.

&lt;syntaxhighlight lang=python&gt;
def checkComplex(x, flag = 0) :
    &quot;&quot;&quot;
status = checkComplex(x [, flag]) 
x must be :
    (v1,v2,'rectangular') or
    (v1,v2,'polar')
    v1,v2 must be type Decimal.
    &quot;&quot;&quot;
    thisName = 'checkComplex (x) :'

    if flag :
        print ()
        print (thisName)
        print ('  Input x =', type(x), (str(x))[:60])

    try:
        error = ''
        v1,v2,str1 = x
        isinstance (v1,dD) or ({}['v1 must be type Decimal.'])
        isinstance (v2,dD) or ({}['v2 must be type Decimal.'])
        (isinstance (str1,str) and str1) or ({}['str1 must be valid string.'])
        str1 = str1.lower()
        if str1 == 'rectangular'[:len(str1)] : pass
        elif str1 == 'polar'[:len(str1)] : pass
        else : ({}['str1 must be &quot;rectangular&quot; or &quot;polar&quot;.'])
    except : error = str(sys.exc_info())
    if error :
        if flag : print_error (error)
        return False
    return True
&lt;/syntaxhighlight&gt;

===str_to_complex (input_string)===

&lt;syntaxhighlight lang=python&gt;
import re

# The re pattern for numbers to be found in string of type complex.
# In string like '  1234567890987654323456E-6 + 987654321234567890987e+2J   '
# extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'.
# eval allows '  12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J   '.
# Conversion to Decimal allows: '__987_65432123____4567890_987___'.
# eval does not allow ' 0123 ' but allows '  2e04 '.
# Conversion to Decimal allows: ' 0123  '
digits  = '[_0123456789]'
integer = '[_0123456789]{1,}'                    # Matches 1 23 345 4567 56789
exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8
Exponent = '({}){{0,1}}'.format(exponent)        # exponent 0 or 1 times.
float1 = '({})\.{{0,1}}'.format(integer)         # Matches 123  2345.
float2 = '{}{{0,}}\.{}'.format(digits, integer)  # Matches   123.345  .345
float_ = '(({})|({}))'.format(float2,float1)
rvalue = '{}({})'.format(float_,Exponent)
rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue)
ivalue_signed = '{}[Jj]'.format(rvalue_signed)


def str_to_complex (input_string) :
    &quot;&quot;&quot;
a,b = str_to_complex (input_string)
input_string could be '   1234567890987654323456E-6 + 987654321234567890987e+2J    '
input_string could be ' ' '  ( 12345678909876543234561234567890987.654323456E-6 +
                             987654321234567890987987654321.23764567890987e+2J  )  ' ' '
or ' ' '
    - (
 (
    12345678_90543123456789054_312345678905431234567890543e-014 +
    543123456789054_3123456789054312345_67890543654327453E-1_3J
) )

    ' ' '
or ' ' '
    - (
 (
    12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_'
    543123456789054_3123456789054312345_67890543654327453E-1_3J
) )

    ' ' '
or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)'
or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)'
or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)',
This function retains precision of a,b
    &quot;&quot;&quot;
    
    try :
        status = 0
        isinstance(input_string,str) or ({}['input_string not type str.'])
        str1 = input_string.strip()
        cx1 = eval(str1)
        isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.'])
    except : status = 1
    if status : return

    # This code removes white lines and comments, if any, at end of each line.
    new_line = '''
'''[-1:]
    lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ]
    lines = [ v for line in lines
                for parts in [ line.split('#') ]
                for v in [ parts[0]] ]
    str1 = ''.join(lines)
    str1 = ''.join(str1.split())

    if isinstance (cx1, (int,float)) :
        resultr = re.search(rvalue_signed, str1)
        resultr or ({}['rvalue not found.'])
        dD1 = dD(resultr[0])
        v1 = eval(str(dD1))
        if v1 == cx1 : return dD1,dD(0)
        if v1 == -cx1 : return -dD1,dD(0)
        ({}['dD1 not recognized.'])

    # cx1 must be complex. It must contain imaginary value.
    resulti = re.search(ivalue_signed, str1)
    resulti or ({}['ivalue not found.'])
    str2j = resulti[0] ; str2 = str2j[:-1]
    dD2 = dD(str2)
    str1 = ''.join(str1.split(str2j))

    # cx1 may contain real value.
    resultr = re.search(rvalue_signed, str1)
    if resultr : dD1 = dD(resultr[0])
    else : dD1 = dD(0)

    cx2 = complex(dD1,dD2)
    if cx2.real == cx1.real : pass
    elif cx2.real == -cx1.real : dD1 = dD1.copy_negate()
    else : ({}['dD1 Not Recognized.'])
    if cx2.imag == cx1.imag : pass
    elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate()
    else : ({}['dD2 Not Recognized.'])

    cx3 = complex(dD1, dD2)
    if cx3 == cx1 : return dD1,dD2
    ({}['No match for cx3.'])

&lt;/syntaxhighlight&gt;

===makeComplex(x)===

&lt;syntaxhighlight lang=python&gt;
def makeComplex (x, flag = 0) :
    '''
result = makeComplex (x[, flag]) 

Input can be tuple with 1,2 or 3 members.
If 1 or 2 members, 'rect' is understood.
The one member or single object may be int, float, complex, CompleX
or string convertible to int, float or complex.

x = makeComplex(4)
x = makeComplex((4,))
x = makeComplex(('4',0))
x = makeComplex((4,'0', 'rect'))
x = makeComplex(4+0j)
x = makeComplex('4+0j')
x = makeComplex(('4+0j',))
In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' )

output is always
(modulus, phase, &quot;polar&quot;) or
(real_part, imag_part, &quot;rect&quot;)
modulus, phase, real_part, imag_part are Decimal objects.

On error returns None.
'''

    thisName = 'makeComplex (x) :'

    if flag :
        print ()
        print (thisName)
        print ('  Input x =', type(x), (str(x))[:60])

    if isinstance(x, CompleX) :  # New class CompleX (note the punctuation.)
        x.check()
        return (x.r, x.i, 'rect')
    
    if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) 

    try :
        status = 1
        a,b = str_to_complex (x) 
    except : status = 0
    if status : return a,b,'rect'

    try :
        status = 1
        result = makeDecimal(x)
        isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.'])
        output = result,dD(0),'rect'
    except : status = 0
    if status : return output

    try :
        status = 1
        v1, = x # Allow for (( 3+4j ),)
    except : status = 0
    if status : return makeComplex (v1)

    try :
        status = 1 ; error = ''
        if len(x) == 2 :
            v1,v2 = [ makeDecimal (v) for v in x  ]
            str1 = 'rect'
        elif len(x) == 3 :
            v1,v2,str1 = x
            v1,v2 = [ makeDecimal (v) for v in (v1,v2)  ]
        else : ({}['len(x) not in (2,3)'])
        output = v1,v2,str1
        checkComplex(output) or ({}['output not valid complex.'])
        if str1[0] in 'rR' : output = v1,v2,'rect'
        else : output = v1,v2,'polar'
    except : status = 0 ; error = str(sys.exc_info())
    if status : return output
    
    if flag : print_error (error)
    return 
&lt;/syntaxhighlight&gt;

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

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of
the section, and also how to calculate the foci and directrices given the equation.
==Latera recta et cetera==
&quot;Latus rectum&quot; is a Latin expression meaning &quot;straight side.&quot;
According to Google, the Latin plural of &quot;latus rectum&quot; is &quot;latera recta,&quot;
but English allows &quot;latus rectums&quot; or possibly &quot;lati rectums.&quot;
The title of this section is poetry to the eyes and music to the ears of a Latin student
and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic.

The translation of the title is &quot;Latus rectums and other things.&quot; This section describes the calculation of interesting items 
associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines.


When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above.
Then verify that the curve is in fact an ellipse.

From these values everything about the ellipse may be calculated. For example:
{{RoundBoxTop|theme=2}}
[[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.&lt;/math&gt;'''
&lt;/br&gt;
&lt;/br&gt;
Axis : (-0.8)x + (-0.6)y + (9.4) = 0&lt;/br&gt;
Eccentricity = 0.9&lt;/br&gt;
&lt;/br&gt;
Directrix 2 : (0.6)x + (-0.8)y + (2) = 0&lt;/br&gt;
Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0&lt;/br&gt;
Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0&lt;/br&gt;
Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0&lt;/br&gt;
Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{ID2}&lt;/math&gt; = (6.32, 7.24)&lt;/br&gt;
&lt;math&gt;\text{I2}&lt;/math&gt;  = (7.204210526315789473684, 6.061052631578947368421)&lt;/br&gt;
F2  = (8, 5)&lt;/br&gt;
M   = (15.16210526315789473684, -4.54947368421052631579)&lt;/br&gt;
F1  = (22.32421052631578947368, -14.09894736842105263158)&lt;/br&gt;
&lt;math&gt;\text{I1}&lt;/math&gt;  = (23.12, -15.16)&lt;/br&gt;
&lt;math&gt;\text{ID1}&lt;/math&gt; = (24.00421052631578947368, -16.33894736842105263158)&lt;/br&gt;
&lt;/br&gt;
P   = (20.30821052631578947368, -15.61094736842105263158)&lt;/br&gt;
Q   = (10.53708406832736953616, -8.018239580333420216299)&lt;/br&gt;
R   = (5.984, 3.488)&lt;/br&gt;
S   = (10.016, 6.512)&lt;/br&gt;
T   = (19.78712645798841993752, -1.080707788087632415281)&lt;/br&gt;
U   = (24.34021052631578947368, -12.58694736842105263158)&lt;/br&gt;
&lt;/br&gt;
Length of major axis: &lt;math&gt;\text{I1I2}&lt;/math&gt; = 26.52631578947368421052&lt;/br&gt;
Length of minor axis: QT = 11.56255298707631300170&lt;/br&gt;
Length of latus rectum: RS = PU = 5.04
]]
Consider conic section: &lt;math&gt;1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.&lt;/math&gt;

This curve is ellipse with random orientation.
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse

result = calculate_abc_epq(ABCDEF)
(len(result) == 2) or 1/0

# ellipse or hyperbola
(abc1,epq1), (abc2,epq2) = result
a1,b1,c1 = abc1 ; e1,p1,q1 = epq1
a2,b2,c2 = abc2 ; e2,p2,q2 = epq2
(e1 == e2) or 2/0
(1 &gt; e1 &gt; 0) or 3/0

print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) )
A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1)
print ('Equation of ellipse in standard form:')
print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0
Equation of ellipse in standard form:
(-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
# python code
def sum_zero(input) :
    '''
sum = sum_zero(input)
If sum is close to 0 and Tolerance permits, sum is returned as 0.
For example: if input contains (2, -1.999999999999999999999)
this function returns sum of these 2 values as 0.
    '''
    global Tolerance
    sump = sumn = 0
    for v in input :
        if v &gt; 0 : sump += v
        elif v &lt; 0 : sumn -= v
    sum = sump - sumn
    if abs(sum) &lt; Tolerance : return (type(Tolerance))(0)
    min, max = sorted((sumn,sump))
    if abs(sum) &lt;= Tolerance*min : return (type(Tolerance))(0)
    return sum
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
===Major axis===
&lt;syntaxhighlight lang=python&gt;
# axis is perpendicular to directrix.
ax,bx = b1,-a1
# axis contains foci. ax + by + c = 0
cx = reduce_Decimal_number(-(ax*p1 + bx*q1))
axis = ax,bx,cx
print ( '    Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) )
print ( '    Eccentricity = {}'.format(e1) )
print ()
print ( '    Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) )
print ( '    Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) )
F1 = p1,q1 # Focus 1.
print ( '    F1 : ({}, {})'.format(p1,q1) )
F2 = p2,q2 # Focus 2.
print ( '    F2 : ({}, {})'.format(p2,q2) )

#  Direction cosines along axis from F1 towards F2:
dx,dy = a1,b1

# p2 = p1 + dx*distance_F1_F2
# q2 = q1 + dy*distance_F1_F2
if dx : distance_F1_F2 = (p2 - p1)/dx
else :  distance_F1_F2 = (q2 - q1)
if distance_F1_F2 &lt; 0 :
    distance_F1_F2 *= -1
    dx *= -1 ; dy *= -1

# Intercept on directrix1
distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1)
ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1
print ( '    Intercept ID1 : ({}, {})'.format(xID1,yID1) )

# 
# distance_F1_F2
# -------------------- = e
# length_of_major_axis
# 
length_of_major_axis = distance_F1_F2 / e1

# Intercept1 on curve
distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2

xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve
I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ]
print ( '    Intercept I1 : ({}, {})'.format(xI1,yI1) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Axis : (-0.8)x + (-0.6)y + (9.4) = 0
    Eccentricity = 0.9

    Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0
    Directrix 2 : (0.6)x + (-0.8)y + (2) = 0
    F1 : (22.32421052631578947368, -14.09894736842105263158)
    F2 : (8, 5)
    Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158)
    Intercept I1 : (23.12, -15.16)
&lt;/syntaxhighlight&gt;
{{RoundBoxTop|theme=2}}
Techniques similar to above can be used to calculate points 
&lt;math&gt;I2, ID2.&lt;/math&gt;
{{RoundBoxBottom}}

===Latus rectums===
&lt;syntaxhighlight lang=python&gt;
# direction cosines along latus rectum.
dlx,dly = -dy,dx
# 
# distance from U to F1               half_latus_rectum
# ------------------------------ = ----------------------- = e1
# distance from U to directrix 1   distance_from_F1_to_ID1
# 
half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1)
# latus rectum 1
# Focal chord has equation (afc)x + (bfc)y + (cfc) = 0.
afc,bfc = a1,b1
cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1))
print ( '    Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) )
        
P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum
print ( '    Point P : ({}, {})'.format(xP,yP) )
U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum
print ( '    Point U : ({}, {})'.format(xU,yU) )
distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt())
print ('    Length PU =', distance)
print ('    half_latus_rectum =', half_latus_rectum)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0
    Point P : (20.30821052631578947368, -15.61094736842105263158)
    Point U : (24.34021052631578947368, -12.58694736842105263158)
    Length PU = 5.04
    half_latus_rectum = 2.52
&lt;/syntaxhighlight&gt;
{{RoundBoxTop|theme=2}}
Techniques similar to above can be used to calculate points 
&lt;math&gt;R, S.&lt;/math&gt;
{{RoundBoxBottom}}

===Minor axis===
&lt;syntaxhighlight lang=python&gt;
print ()

# Mid point between F1, F2:
M = xM,yM = (p1 + p2)/2, (q1 + q2)/2
print ( '    Mid point M : ({}, {})'.format(xM,yM) )
half_major  = length_of_major_axis / 2
half_distance = distance_F1_F2 / 2

# half_distance**2 + half_minor**2 = half_major**2
half_minor = ( half_major**2 - half_distance**2 ).sqrt()
length_of_minor_axis = half_minor * 2
        
Q = xQ,yQ = xM + dlx*half_minor,  yM + dly*half_minor
T = xT,yT = xM - dlx*half_minor,  yM - dly*half_minor
print ( '    Point Q : ({}, {})'.format(xQ,yQ) )
print ( '    Point T : ({}, {})'.format(xT,yT) )

print ('    length_of_major_axis =', length_of_major_axis)
print ('    length_of_minor_axis =', length_of_minor_axis)
# 
# A basic check.
# length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2)
# 
# length_of_minor_axis**2
# ----------------------- = 1-e**2
# length_of_major_axis**2
# 
# length_of_minor_axis**2
# ----------------------- + (e**2 - 1) = 0
# length_of_major_axis**2
# 
values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1
sum_zero(values) and 3/0
        
aM,bM = a1,b1 # Minor axis is parallel to directrix.
cM = reduce_Decimal_number(-(aM*xM + bM*yM))
print ( '    Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Mid point M : (15.16210526315789473684, -4.54947368421052631579)
    Point Q : (10.53708406832736953616, -8.018239580333420216299)
    Point T : (19.78712645798841993752, -1.080707788087632415281)
    length_of_major_axis = 26.52631578947368421052
    length_of_minor_axis = 11.56255298707631300170
    Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

===Checking===
{{RoundBoxTop|theme=2}}
All interesting points have been calculated without using equations of any of the relevant lines.

However, equations of relevant lines are very useful for testing, for example:


* Check that points &lt;math&gt;ID2, I2, F2, M, F1, I1, ID1&lt;/math&gt; are on axis.

* Check that points &lt;math&gt;R, F2, S&lt;/math&gt; are on latus rectum through &lt;math&gt;F2.&lt;/math&gt;

* Check that points &lt;math&gt;Q, M, T&lt;/math&gt; are on minor axis through &lt;math&gt;M.&lt;/math&gt;

* Check that points &lt;math&gt;P, F1, U&lt;/math&gt; are on latus rectum through &lt;math&gt;F1.&lt;/math&gt;


Test below checks that 8 points &lt;math&gt;I1, I2, P, Q, R, S, T, U&lt;/math&gt; are on ellipse and satisfy eccentricity &lt;math&gt;e = 0.9.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
t1 = (
    ('I1'), ('I2'),
    ('P'), ('Q'), ('R'),
    ('S'), ('T'), ('U'),
)

for name in t1 :
    value = eval(name)
    x,y = [ reduce_Decimal_number(v) for v in value ]
    print ('{} : ({}, {})'.format((name+' ')[:2], x,y))
    values = A*x**2, B*y**2, C*x*y, D*x, E*y, F
    sum_zero(values) and 3/0

    # Relative to Directrix 1 and Focus 1:
    distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt()
    distance_to_directrix1 = a1*x + b1*y + c1
    e1 = distance_to_F1 / distance_to_directrix1
    print ('    e1 =',e1) # Raw value is printed.

    # Relative to Directrix 2 and Focus 2:
    distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt()
    distance_to_directrix2 = a2*x + b2*y + c2
    e2 = distance_to_F2 / distance_to_directrix2
    e2 = reduce_Decimal_number(e2)
    print ('    e2 =',e2) # Clean value is printed.
&lt;/syntaxhighlight&gt;
{{RoundBoxTop|theme=2}}
Note the differences between &quot;raw&quot; values of &lt;math&gt;e_1&lt;/math&gt; and &quot;clean&quot; values of &lt;math&gt;e_2.&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
I1 : (23.12, -15.16)
    e1 = -0.9000000000000000000034 
    e2 = 0.9
I2 : (7.204210526315789473684, 6.061052631578947368421)
    e1 = -0.9 
    e2 = 0.9
P  : (20.30821052631578947368, -15.61094736842105263158) 
    e1 = -0.9 
    e2 = 0.9
Q  : (10.53708406832736953616, -8.018239580333420216299)
    e1 = -0.9000000000000000000002
    e2 = 0.9
R  : (5.984, 3.488)
    e1 = -0.9000000000000000000003
    e2 = 0.9
S  : (10.016, 6.512) 
    e1 = -0.9000000000000000000003
    e2 = 0.9   
T  : (19.78712645798841993752, -1.080707788087632415281)
    e1 = -0.8999999999999999999996
    e2 = 0.9
U  : (24.34021052631578947368, -12.58694736842105263158)
    e1 = -0.9
    e2 = 0.9
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

==Traditional definition of ellipse==
{{RoundBoxTop|theme=2}}
[[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.'''
&lt;/br&gt;
Sum of distance &lt;math&gt;F_1G&lt;/math&gt; and distance &lt;math&gt;F_2G&lt;/math&gt; is constant.
]]
Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant.


In the diagram the two fixed points are the foci, Focus 1 or &lt;math&gt;F_1&lt;/math&gt; and Focus 2 or &lt;math&gt;F_2.&lt;/math&gt;

Distance between &lt;math&gt;F_1&lt;/math&gt; and &lt;math&gt;F_2&lt;/math&gt;, distance &lt;math&gt;F_1F_2&lt;/math&gt;, must be non-zero.

Point &lt;math&gt;G&lt;/math&gt; on perimeter of ellipse moves so that sum of distance &lt;math&gt;F_1G&lt;/math&gt;
and distance &lt;math&gt;F_2G&lt;/math&gt; is constant.


Points &lt;math&gt;T_1&lt;/math&gt; and &lt;math&gt;T_2&lt;/math&gt; are on axis of ellipse and the same rule applies to these points.

distance &lt;math&gt;F_1T_1&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt; is constant.


distance &lt;math&gt;F_1T_1&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt;

&lt;math&gt;=&lt;/math&gt; distance &lt;math&gt;F_1G&lt;/math&gt; + distance &lt;math&gt;F_2G&lt;/math&gt;

&lt;math&gt;=&lt;/math&gt; distance &lt;math&gt;F_2T_2&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt;

&lt;math&gt;= \text{length of major axis.}&lt;/math&gt;

Therefore the constant is &lt;math&gt;\text{length of major axis}&lt;/math&gt; which must be greater than distance &lt;math&gt;F_1F_2.&lt;/math&gt;


From information given, calculate eccentricity &lt;math&gt;e&lt;/math&gt; and equation of one directrix.
Choose directrix 1 &lt;math&gt;dx1&lt;/math&gt; associated with focus F1.


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
==Ellipse at origin==
{{RoundBoxTop|theme=2}}
[[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.'''
&lt;/br&gt;
Sum of distance &lt;math&gt;F_1P&lt;/math&gt; and distance &lt;math&gt;F_2P&lt;/math&gt; is constant.
]]
Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant.


By definition distance &lt;math&gt;F_2P&lt;/math&gt; + distance &lt;math&gt;F_1P&lt;/math&gt; is constant.

&lt;math&gt;\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)&lt;/math&gt;

Expand &lt;math&gt;(1)&lt;/math&gt; and result is &lt;math&gt;Ax^2 + By^2 + F = 0\ \dots\ (2)&lt;/math&gt; where:

&lt;math&gt;A = 4k^2 - 16p^2&lt;/math&gt;

&lt;math&gt;B = 4k^2&lt;/math&gt;

&lt;math&gt;F = 4k^2p^2 - k^4&lt;/math&gt;


When &lt;math&gt;y = 0,&lt;/math&gt; point &lt;math&gt;B,\ Ax^2 = -F&lt;/math&gt;

&lt;math&gt;x^2 = \frac{-F}{A}&lt;/math&gt;
&lt;math&gt;= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}&lt;/math&gt;
&lt;math&gt;=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.&lt;/math&gt;

Therefore: &lt;math&gt;x = \frac{k}{2} = a&lt;/math&gt;

&lt;math&gt;k = \text{length of major axis.}&lt;/math&gt;

By definition, distance &lt;math&gt;F_2A&lt;/math&gt; + distance &lt;math&gt;F_1A = k.&lt;/math&gt;

Therefore distance &lt;math&gt;F_1A = a.&lt;/math&gt;


Intercept form of ellipse at origin:

&lt;math&gt;(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2&lt;/math&gt;

&lt;math&gt;\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1&lt;/math&gt;

&lt;math&gt;\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1&lt;/math&gt;

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

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

==Second definition of ellipse==
{{RoundBoxTop|theme=2}}
[[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt; where &lt;math&gt;a,b = 20,12&lt;/math&gt;.'''
&lt;/br&gt;
At point &lt;math&gt;B,\ \frac{u}{v} = e.&lt;/math&gt;
&lt;/br&gt;
At point &lt;math&gt;A,\ \frac{a}{t} = e.&lt;/math&gt;
]]
Ellipse is path of point that moves so that ratio of distance to
fixed point and distance to fixed line is constant.


Let &lt;math&gt;\frac{p}{a} = e&lt;/math&gt; where:

* &lt;math&gt;p&lt;/math&gt; is non-zero,

* &lt;math&gt;a &gt; p,&lt;/math&gt;

* &lt;math&gt;a = p + u.&lt;/math&gt;

Therefore, &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;

Let directrix have equation &lt;math&gt;x = t&lt;/math&gt; where &lt;math&gt;\frac{a}{t} = e.&lt;/math&gt;

At point &lt;math&gt;B:&lt;/math&gt;

&lt;math&gt;\frac{p}{p+u} = \frac{p+u}{p+u+v} = e&lt;/math&gt;

&lt;math&gt;(p+u)^2 = p(p+u+v)&lt;/math&gt;

&lt;math&gt;pp + pu + pu + uu = pp + pu + pv&lt;/math&gt;

&lt;math&gt;pu + uu = pv&lt;/math&gt;

&lt;math&gt;u(p + u) = pv&lt;/math&gt;

&lt;math&gt;\frac{u}{v} = \frac{p}{p+u} = e&lt;/math&gt;

&lt;math&gt;\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)&lt;/math&gt;

Statement &lt;math&gt;(3)&lt;/math&gt; is true at point &lt;math&gt;A&lt;/math&gt; also.


Section under &quot;Proof&quot; below proves that statement (3) is true for any point &lt;math&gt;P&lt;/math&gt; on ellipse.
{{RoundBoxBottom}}
===Proof===
{{RoundBoxTop|theme=2}}
[[File:0902ellipse00.png|thumb|400px|'''Proving that &lt;math&gt;\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e&lt;/math&gt;.'''
&lt;/br&gt;
Graph is part of curve &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.&lt;/math&gt;
&lt;/br&gt;
distance to Directrix1 &lt;math&gt;= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.&lt;/math&gt;
&lt;/br&gt;
base = &lt;math&gt;x - p = x - ae&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{(distance to Focus1)}^2 = \text{base}^2 + y^2&lt;/math&gt;
]]
As expressed above in statement &lt;math&gt;3,&lt;/math&gt; second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to
fixed point and distance to fixed line is constant.


This section proves that this definition is true for any point &lt;math&gt;P&lt;/math&gt; on the ellipse.


At point &lt;math&gt;P:&lt;/math&gt;

&lt;math&gt;(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0&lt;/math&gt;

&lt;math&gt;y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}&lt;/math&gt;

&lt;math&gt;= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}&lt;/math&gt;

&lt;math&gt;= e^2x^2 - x^2 + a^2 - a^2e^2&lt;/math&gt;

base &lt;math&gt;= x-p = x-ae&lt;/math&gt;

&lt;math&gt;(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2&lt;/math&gt;
&lt;math&gt;= a^2 - 2aex + e^2x^2&lt;/math&gt;
&lt;math&gt;= (a-ex)^2&lt;/math&gt;

&lt;math&gt;\text{distance to Focus1} = \text{distance}\ F_1P = a - ex&lt;/math&gt;

&lt;math&gt;\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}&lt;/math&gt;

&lt;math&gt;\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}&lt;/math&gt;
&lt;math&gt;= (a - ex)\frac{e}{(a-ex)}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;


Similar calculations can be used to prove the case for Focus2 &lt;math&gt;(-p, 0)&lt;/math&gt; and Directrix2 &lt;math&gt;(x = -t)&lt;/math&gt;
in which case:

&lt;math&gt;\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}&lt;/math&gt;
&lt;math&gt;= (a + ex)\frac{e}{(a + ex)}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;

Therefore: &lt;math&gt;\frac{\text{distance to focus}}{\text{distance to directrix}} = e&lt;/math&gt; where &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;


Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity &lt;math&gt;e.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

==Heading==
===Properties of ellipse===
{{RoundBoxTop|theme=2}}
[[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.'''
&lt;/br&gt;
&lt;/br&gt;
Traditional definition of ellipse:
&lt;/br&gt;
&lt;math&gt;\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Second definition of ellipse:
&lt;/br&gt;
&lt;math&gt;\frac{\text{distance } AF_1}  {\text{distance } AG } = \text{eccentricity } e.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Triangle &lt;math&gt;A F_1  G&lt;/math&gt; is right triangle.
&lt;/br&gt;
&lt;math&gt;e = \cos \angle  O F_1  A  = \cos \angle   F_1 A G&lt;/math&gt;
]]
Ellipse in diagram has:


* Two foci: &lt;math&gt;F_1\ (p,0),\ F_2\ (-p,0).&lt;/math&gt;

* Length of major axis &lt;math&gt;= \text{distance } I_2 I_1 = 2a&lt;/math&gt;

* Length of minor axis &lt;math&gt;= \text{distance } A B = 2b&lt;/math&gt;

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

* Length of latus rectum &lt;math&gt;= \text{distance } P Q&lt;/math&gt;

* Distance between directrices &lt;math&gt;= \text{distance } D_2 D_1 = 2t&lt;/math&gt;


Properties of ellipse:

* &lt;math&gt;\frac{\text{length of major axis}}  {\text{distance between directrices}} = e&lt;/math&gt;


* &lt;math&gt;\frac{\text{distance between foci}}  {\text{length of major axis}} = e&lt;/math&gt;


* &lt;math&gt;\frac{\text{distance between foci}}  {\text{distance between directrices}}= e^2&lt;/math&gt;


* &lt;math&gt;(\frac{\text{length of minor axis}}  {\text{length of major axis}})^2 + e^2 = 1&lt;/math&gt;


* &lt;math&gt;\frac{\text{length of latus rectum}}  {\text{length of major axis}} + e^2 = 1&lt;/math&gt;


* line &lt;math&gt;PD_1&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
====Major axis====
From traditional definition of ellipse:

Distance &lt;math&gt;AF_2\ +&lt;/math&gt; distance &lt;math&gt;AF_1&lt;/math&gt;

= distance &lt;math&gt;I_1F_1\ +&lt;/math&gt; distance &lt;math&gt;I_1F_2&lt;/math&gt;

= distance &lt;math&gt;I_2F_2\ +&lt;/math&gt; distance &lt;math&gt;I_2F_1&lt;/math&gt;

= &lt;math&gt;k.&lt;/math&gt;


Therefore: 

Length of major axis = distance &lt;math&gt;I_2I_1 = 2a = k.&lt;/math&gt;

Distance &lt;math&gt;AF_1 = \frac{k}{2} = a.&lt;/math&gt;


From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e&lt;/math&gt;
&lt;math&gt;= \frac{\text{distance }OI_1}{\text{distance }OD_1}.&lt;/math&gt;

&lt;math&gt;\frac{\text{length of major axis}}{\text{distance between directrices}} = e.&lt;/math&gt;
====Foci====
From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.&lt;/math&gt;

&lt;math&gt;a - p = te - ae&lt;/math&gt;

&lt;math&gt;a - p = a - ae&lt;/math&gt;

Therefore: &lt;math&gt;p = ae&lt;/math&gt; or &lt;math&gt;\frac{p}{a} = e.&lt;/math&gt;

&lt;math&gt;\frac{\text{distance between foci}}{\text{length of major axis}} = e.&lt;/math&gt;

&lt;math&gt;\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.&lt;/math&gt;
====Minor axis====
Triangle &lt;math&gt;AOF_1&lt;/math&gt; is right triangle.

&lt;math&gt;\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1&lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    (\frac{p}{a})^2  &lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    (\frac{ae}{a})^2  &lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    e^2  = 1&lt;/math&gt;


&lt;math&gt;( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1&lt;/math&gt;


Triangles &lt;math&gt;AOF_1,\ AF_1G&lt;/math&gt; are similar.

Triangle &lt;math&gt;AF_1G&lt;/math&gt; is right triangle.


&lt;math&gt;e = \cos \angle OF_1A = \cos \angle F_1AG.&lt;/math&gt;
====Latus rectum====
From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e&lt;/math&gt;

&lt;math&gt;\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).&lt;/math&gt;

&lt;math&gt;\frac{\text{distance }PF_1} {a} = 1 - e^2.&lt;/math&gt;

&lt;math&gt; \frac{\text{length of latus rectum}} {\text{length of major axis}}  + e^2 = 1&lt;/math&gt;

====Slope of curve====
Curve has equation: &lt;math&gt;b^2x^2 + a^2y^2 = a^2b^2&lt;/math&gt;

Derivative of both sides:

&lt;math&gt;b^22x + a^22yy' = 0&lt;/math&gt;

&lt;math&gt;y' = \frac{-xb^2}{ya^2}&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;math&gt;= \frac{-x(1-e^2)}{y}&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

At point &lt;math&gt;P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}&lt;/math&gt;
&lt;math&gt;= \frac{ae}{a} = e.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

Slope of line &lt;math&gt;PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

&lt;math&gt;m_1 = m_2.&lt;/math&gt; Therefore line &lt;math&gt;PD_1&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Intercept form of equation===


&lt;math&gt;\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.'''
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
Equation of ellipse has format &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt; where:
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{Length of minor axis} = 2b = \text{distance}\ BA = 24&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
]]
In diagram:

Intercept &lt;math&gt;I_1&lt;/math&gt; has coordinates &lt;math&gt;(a,0).&lt;/math&gt;

Intercept &lt;math&gt;I_2&lt;/math&gt; has coordinates &lt;math&gt;(-a,0).&lt;/math&gt;

Intercept &lt;math&gt;A&lt;/math&gt; has coordinates &lt;math&gt;(0,b).&lt;/math&gt;

Intercept &lt;math&gt;B&lt;/math&gt; has coordinates &lt;math&gt;(0,-b).&lt;/math&gt;

Focus &lt;math&gt;F_1&lt;/math&gt; has coordinates &lt;math&gt;(f,0)&lt;/math&gt; where &lt;math&gt;f = ea.&lt;/math&gt;

Focus &lt;math&gt;F_2&lt;/math&gt; has coordinates &lt;math&gt;(-f,0).&lt;/math&gt;

Curve has equation &lt;math&gt;\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,&lt;/math&gt; called intercept form of equation of ellipse because intercepts are apparent as the 
fractional value of each coefficient.


Standard form of this equation is: &lt;math&gt;(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.&lt;/math&gt; While the standard form is
valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and
intercepts are neatly contained within the equation.


Slope of curve:

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

Derivative of both sides:

&lt;math&gt;b^22x + a^22yy' = 0&lt;/math&gt;

&lt;math&gt;y' = \frac{-xb^2}{ya^2}&lt;/math&gt;
&lt;math&gt;= \frac{-x(1-e^2)}{y}&lt;/math&gt;


At point &lt;math&gt;P&lt;/math&gt; on latus rectum &lt;math&gt;PQ:&lt;/math&gt;

&lt;math&gt;m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e&lt;/math&gt;


Slope of line &lt;math&gt;PD = m_2 = \frac{PF_1}{F_1D} = e&lt;/math&gt;

&lt;math&gt;m_1 = m_2.&lt;/math&gt; Line &lt;math&gt;PD&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

===Example===
{{RoundBoxTop|theme=2}}
[[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.'''
&lt;/br&gt;
M is mid-point between F1 and F2.
&lt;/br&gt;
Point R is on minor axis.
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{F1:}\ (1, -7)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{F2:}\ (-1.24, 0.68)&lt;/math&gt;
&lt;/br&gt;
length_of_major_axis = 10
&lt;/br&gt;
&lt;math&gt;\text{M:}\ (-0.12, -3.16)&lt;/math&gt;
&lt;/br&gt;
length_of_minor_axis = 6
&lt;/br&gt;
&lt;math&gt;\text{R:}\ (2.76, -2.32)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e = 0.8&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{D1:}\ (1.63, -9.16)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{epq}\ =\ (0.8,\ 1,\ -7)&lt;/math&gt;
]]
Given:
&lt;syntaxhighlight lang=python&gt;
# python code

F1 = 1, -7          # Focus 1
F2 = -1.24, 0.68    # Focus 2
length_of_major_axis = 10
&lt;/syntaxhighlight&gt;
Calculate equation of ellipse.
&lt;syntaxhighlight lang=python&gt;

F1 = p1,q1 = [ dD(str(v)) for v in F1 ]    # Focus 1
F2 = p2,q2 = [ dD(str(v)) for v in F2 ]    # Focus 2

length_of_major_axis = dD(length_of_major_axis)
half_major_axis = length_of_major_axis / 2

# Direction cosines from F1 to F2
dx = p2-p1 ; dy = q2-q1
divider = (dx**2 + dy**2).sqrt()
dx,dy = [ (v/divider) for v in (dx,dy) ]

# F2
# p2 = p1 + dx*distance_F1_F2
# q2 = q1 + dy*distance_F1_F2
distance_F1_F2 = (q2-q1)/dy
half_distance_F1_F2 = distance_F1_F2 / 2

# The mid-point
M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2

# Eccentricity:
e = distance_F1_F2 / length_of_major_axis

# distance from point R to F1                half_major_axis
# ------------------------------------ = e = -----------------------------------------
# distance from point R to Directrix 1       distance from point M to Directrix 1

distance_from_point_M_to_dx1 = half_major_axis / e

# Intersection of axis and directrix 1
D1 = xM-dx*distance_from_point_M_to_dx1,  yM-dy*distance_from_point_M_to_dx1
D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ]

# Equation of Directrix 1
# dx1 = adx1,bdx1,cdx1
adx1,bdx1 = dx, dy # Perpendicular to axis.
# adx1*x + bdx1*y + cdx1 = 0
# Directrix 1 contains point D1
cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) )

abc = adx1,bdx1,cdx1
epq = e,p1,q1
ABCDEF = ABCDEF_from_abc_epq  (abc,epq, 1)
&lt;/syntaxhighlight&gt;
Equation of ellipse in standard form:

&lt;math&gt;(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0&lt;/math&gt;


For more insight into method of calculation and proof:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    print ('F1: ({}, {})'.format(p1,q1))
    print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1))

    print ('F2: ({}, {})'.format(p2,q2))
    print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2))

    print ('length_of_major_axis =', length_of_major_axis)
    
    print ('M: ({}, {})'.format(xM,yM))
    print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM))

    # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2
    half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt()
    length_of_minor_axis = half_minor_axis * 2
    s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1))

    # Direction cosines on major axis:
    print ('dx,dy =', dx,dy)
    
    # Direction cosines on minor axis:
    dnx,dny = dy,-dx
    print ('dnx,dny =', dnx,dny)

    # One point on minor axis:
    R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis
    print ('R: ({}, {})'.format(xR,yR))
    print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR))
    # Verify that point R is on ellipse:
    sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0

    s1 = 'e' ; print (s1, '=', eval(s1))

    print ('D1: ({}, {})'.format(xD1,yD1))
    print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1))

    print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1))

    print()
# For proof, reverse the process:    
    (abc1,epq1), (abc2,epq2) = calculate_abc_epq  (ABCDEF)
    a1,b1,c1 = abc1 ; e1,p1,q1 = epq1
    print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1))
    print ('Eccentricity e1: {}'.format(e1))
    print ('F1: ({}, {})'.format(p1,q1))

    print()
    a2,b2,c2 = abc2 ; e2,p2,q2 = epq2
    print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2))
    print ('Eccentricity e2: {}'.format(e2))
    print ('F2: ({}, {})'.format(p2,q2))

    print ('\nEquation of ellipse with integer coefficients:')
    A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ]
    str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'
    print (str1.format(A,B,C,D,E,F))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
F1: (1, -7)
F1: (x - (1))^2 + (y - (-7))^2 = 1
F2: (-1.24, 0.68)
F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1
length_of_major_axis = 10
M: (-0.12, -3.16)
M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1
length_of_minor_axis = 6
dx,dy = -0.28 0.96
dnx,dny = 0.96 0.28
R: (2.76, -2.32)
R: (x - (2.76))^2 + (y - (-2.32))^2 = 1
e = 0.8
D1: (1.63, -9.16)
D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1
Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0

Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0
Eccentricity e1: 0.8
F1: (1, -7)

Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 
Eccentricity e2: 0.8
F2: (-1.24, 0.68) 

Equation of ellipse with integer coefficients: 
&lt;/syntaxhighlight&gt;
&lt;math&gt;(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

=allEqual=

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;
{| class=&quot;wikitable&quot;
|-
! &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2 - N&lt;/math&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;6&lt;/code&gt; || &lt;code&gt;-221&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;7&lt;/code&gt; || &lt;code&gt;-208&lt;/code&gt;
|-
|-
| &lt;code&gt;10&lt;/code&gt; || &lt;code&gt;-157&lt;/code&gt;
|-
| &lt;code&gt;11&lt;/code&gt; || &lt;code&gt;-136&lt;/code&gt;
|-
| &lt;code&gt;12&lt;/code&gt; || &lt;code&gt;-113&lt;/code&gt;
|-
| &lt;code&gt;13&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-88&lt;/code&gt;
|-
| &lt;code&gt;26&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;419&lt;/code&gt;
|}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
&lt;/math&gt;</text>
      <sha1>7v48256re8rlvwwl5muohsgn80au6b0</sha1>
    </revision>
    <revision>
      <id>2815178</id>
      <parentid>2815177</parentid>
      <timestamp>2026-06-11T08:58:28Z</timestamp>
      <contributor>
        <username>ThaniosAkro</username>
        <id>2805358</id>
      </contributor>
      <comment>/* Implementation */</comment>
      <origin>2815178</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="56560" sha1="53pvdg4yicv3c1rtj18paru7auw2kun" xml:space="preserve">

==Table==

{| class=&quot;wikitable&quot;
|-
| From &lt;math&gt;(3):&lt;/math&gt; || &lt;math&gt;p^3&lt;/math&gt;|| || &lt;math&gt;+Ap&lt;/math&gt;  || &lt;math&gt;+B&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (5)&lt;/math&gt;
|-
| From &lt;math&gt;(4):&lt;/math&gt; || || &lt;math&gt;Cp^2&lt;/math&gt;|| &lt;math&gt;&lt;/math&gt;  || &lt;math&gt;+D&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (6)&lt;/math&gt;
|-
| &lt;math&gt;(5)*D&lt;/math&gt; || &lt;math&gt;Dp^3&lt;/math&gt;|| || &lt;math&gt;+DAp&lt;/math&gt;  || &lt;math&gt;+DB&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (7)&lt;/math&gt;
|-
| &lt;math&gt;(6)*B&lt;/math&gt; || || &lt;math&gt;BCp^2&lt;/math&gt;|| &lt;math&gt;&lt;/math&gt;  || &lt;math&gt;+BD&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (8)&lt;/math&gt;
|-
| &lt;math&gt;(7)-(8)&lt;/math&gt; || &lt;math&gt;Dp^3&lt;/math&gt;||&lt;math&gt;-BCp^2&lt;/math&gt; || &lt;math&gt;+DAp&lt;/math&gt;  || &lt;math&gt;&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (9)&lt;/math&gt;
|-
| Simplify &lt;math&gt;(9)&lt;/math&gt; || &lt;math&gt;&lt;/math&gt;||&lt;math&gt;Dp^2&lt;/math&gt; || &lt;math&gt;-BCp&lt;/math&gt;  || &lt;math&gt;+DA&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (10)&lt;/math&gt;
|-
| &lt;math&gt;(6)*A&lt;/math&gt; || || &lt;math&gt;ACp^2&lt;/math&gt;|| &lt;math&gt;&lt;/math&gt;  || &lt;math&gt;+AD&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (11)&lt;/math&gt;
|-
| &lt;math&gt;(10)-(11)&lt;/math&gt; || &lt;math&gt;&lt;/math&gt;||&lt;math&gt;Dp^2-ACp^2&lt;/math&gt; || &lt;math&gt;-BCp&lt;/math&gt;  || &lt;math&gt;&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (12)&lt;/math&gt;
|-
| Simplify &lt;math&gt;(12)&lt;/math&gt; || &lt;math&gt;&lt;/math&gt;|| || &lt;math&gt;Dp-ACp&lt;/math&gt; || &lt;math&gt;-BC&lt;/math&gt;  || &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (13)&lt;/math&gt;
|}

==Implementation==

&lt;math&gt;(1)&lt;/math&gt; squared: &lt;math&gt;a^2 = p^6 - 6p^4q^2\ + 9p^2q^4&lt;/math&gt;

&lt;math&gt;p^2 p^2 p^2 - (6) p^2 p^2 q^2\ + (9)p^2q^4 - a^2\dots\ (1a)&lt;/math&gt;


&lt;math&gt;(1a) * (3q)^3:&lt;/math&gt;

&lt;math&gt;3qp^2\ 3qp^2\ 3qp^2 - (6) 3q\ 3qp^2\ 3qp^2\ q^2\ + (9)3q3q\ 3qp^2\ q^4 - 3q3q3qa^2\dots\ (1b)&lt;/math&gt;

From &lt;math&gt;(2):\ 3qp^2 = b + q^3\ \dots\ (2a)&lt;/math&gt;


For &lt;math&gt;(3qp^2)&lt;/math&gt; in &lt;math&gt;(1b)&lt;/math&gt; substitute &lt;math&gt; (b + q^3) : &lt;/math&gt;

&lt;math&gt;(b + q^3) (b + q^3) (b + q^3) - (6) 3q (b + q^3) (b + q^3) q^2\ + (9)3q3q (b + q^3) q^4 - 3q3q3qa^2\dots\ (1c)&lt;/math&gt;

==Complex Cube Root==

See [https://en.wikiversity.org/wiki/Cubic_function#Vieta's_substitution Vieta's Substitution.]

The depressed cubic function is : &lt;math&gt;f(t) = t^3 + At + B = 0.&lt;/math&gt;

Let &lt;math&gt;A = -3C&lt;/math&gt; and let &lt;math&gt;t = w + \frac{C}{w} = \frac{w^2 + C}{w}.&lt;/math&gt;

Substitute for &lt;math&gt;A, t&lt;/math&gt; in the depressed function and result is:

&lt;math&gt;f(w) = w^6 + Bw^3 + C^3&lt;/math&gt; or

&lt;math&gt;f(W) = W^2 + BW + C^3&lt;/math&gt; where &lt;math&gt;W = w^3&lt;/math&gt; and &lt;math&gt;w = \sqrt[3]{W}&lt;/math&gt;.

From the quadratic formula: &lt;math&gt;W = \frac{-B \pm \sqrt{B^2 - 4C^3}}{2} = \frac{-B \pm \sqrt{4C^3 - B^2}\sqrt{-1}   }{2}&lt;/math&gt;

However, &lt;math&gt;W = a + bi.&lt;/math&gt;

Therefore: &lt;math&gt;a = \frac{-B}{2}&lt;/math&gt; and &lt;math&gt;b = \frac{ \sqrt{4C^3 - B^2} }{2}.&lt;/math&gt;

&lt;math&gt;B = -2a&lt;/math&gt; and

&lt;math&gt;2b = \sqrt{4C^3 - B^2} &lt;/math&gt;

&lt;math&gt;4b^2 = 4C^3 - B^2 &lt;/math&gt;

&lt;math&gt;4C^3 = 4b^2 + B^2 &lt;/math&gt;

&lt;math&gt;C^3 = b^2 + \frac{B^2}{4} &lt;/math&gt;

&lt;math&gt;C = \sqrt[3]{C^3} &lt;/math&gt; and

&lt;math&gt;A = -3C. &lt;/math&gt;

&lt;math&gt;A,B &lt;/math&gt; of &lt;math&gt;f(t)&lt;/math&gt; have been defined with &lt;math&gt;C&lt;/math&gt; positive and &lt;math&gt;A&lt;/math&gt; negative.

Because &lt;math&gt;W&lt;/math&gt; has three complex roots, &lt;math&gt;f(t)&lt;/math&gt; must have three real roots.

Calculate one of the roots of &lt;math&gt;f(t).&lt;/math&gt;

&lt;math&gt;t = \frac{w^2 + C}{w}.&lt;/math&gt;

Therefore &lt;math&gt;wt = w^2 + C&lt;/math&gt; and

&lt;math&gt;w^2 - tw + C = 0.&lt;/math&gt;

From the quadratic formula : 

&lt;math&gt;w = \frac{t \pm \sqrt{t^2 - 4C}}{2} = \frac{t \pm \sqrt{4C - t^2}\sqrt{-1}   }{2}&lt;/math&gt;

&lt;math&gt;w = p + qi&lt;/math&gt;

Therefore: &lt;math&gt;p = \frac{t}{2}&lt;/math&gt; and &lt;math&gt;Q = \frac{4C - t^2  }{4} = C - \frac{t^2}{4}&lt;/math&gt; where &lt;math&gt;Q = q^2.&lt;/math&gt;

&lt;math&gt;q = \sqrt{Q}&lt;/math&gt; but sign of this calculation of &lt;math&gt;q&lt;/math&gt; is ambiguous.

Let &lt;math&gt;P = p^2&lt;/math&gt;

&lt;math&gt;q = \frac{b}{(3P - Q)}&lt;/math&gt;

and &lt;math&gt;(p + qi)^3 = a + bi.&lt;/math&gt;
===An Example===

Calculate cube roots of complex number &lt;math&gt;W = -39582 + 3799i.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:1215depressed cubic01.png|thumb|400px|'''Graph of &lt;math&gt;f(t)&lt;/math&gt; shown as graph of &lt;math&gt;f(x)&lt;/math&gt; and showing three values of &lt;math&gt;t: t_1, t_2, t_3&lt;/math&gt;.'''
&lt;/br&gt;
&lt;math&gt;Y&lt;/math&gt; axis compressed for clarity.
&lt;/br&gt;
&lt;math&gt;A,B = -3495.0, 79164.0&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;t_1 = -68.229473419497441512295943903670298\dots&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;t_2 = 32.229473419497441512295943903670298\dots&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\ t_3 = 36&lt;/math&gt;
]]

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

import decimal
decimal.getcontext().prec = 22
ab = -39582,3799
a,b = [ dD(v) for v in ab ]
B = -2*a
C = (b**2 + B*B/4) ** (dD(1)/3)
A = -3*C

a,b,A,B,C = [ dD(str(float(v))) for v in (a,b,A,B,C) ]
print ( 'a,b = {}, {}'.format(a,b) )
print ( 'A,B,C = {}, {}, {}'.format(A,B,C) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
a,b = -39582.0, 3799.0
A,B,C = -3495.0, 79164.0, 1165.0
&lt;/syntaxhighlight&gt;
Calculate roots of cubic function:

&lt;math&gt;y = f (t) &lt;/math&gt;&lt;math&gt; = t^3 &lt;/math&gt;&lt;math&gt; - 3495 t &lt;/math&gt;&lt;math&gt; + 79164 .&lt;/math&gt;

Three roots are: &lt;math&gt;t_1,\ t_2,\ t_3 = -68.22947341949744\dots,\ 32.22947341949744\dots,\ 36.0&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
# python code:
t1, t2, t3 = ('-68.229473419497441512295943903670298',
               '32.229473419497441512295943903670298', 36 )
values_of_t = [ dD(v) for v in (t1,t2,t3) ]

for t in values_of_t :
    p = t/2 ; P = p**2
    Q = C - t*t/4 ; q = b/(3*P - Q)
    # Check results:
    print ()
    sx = 't' ; print (sx,'=', eval(sx))
    print ( 'p,q = {}, {}'.format(p,q) )
    ab = [ p*P - 3*p*Q, 3*P*q - q*Q ]
    a_,b_ = [ float(v) for v in ab ]
    sx = 'a_,b_' ; print (sx,'=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
t = -68.229473419497441512295943903670298
p,q = -34.11473670974872075615, 1.088457268119895641747
a_,b_ = (-39582.0, 3799.0)
    
t = 32.229473419497441512295943903670298
p,q = 16.11473670974872075615, -30.08845726811989564176
a_,b_ = (-39582.0, 3799.0)
    
t = 36
p,q = 18, 29
a_,b_ = (-39582.0, 3799.0)
&lt;/syntaxhighlight&gt;
Three cube roots of &lt;math&gt;W = -39582 + 3799i&lt;/math&gt; are:

&lt;math&gt;w_1 = -34.11473670974872075615 + 1.088457268119895641747i&lt;/math&gt;

&lt;math&gt;w_2 = 16.11473670974872075615 - 30.08845726811989564176i&lt;/math&gt;

&lt;math&gt;w_3 = 18 + 29i&lt;/math&gt;

==Making the Decimal object==

The following function verifies that we are working with Decimal objects. 

&lt;syntaxhighlight lang=python&gt;
import sys
import decimal

getcontext = decimal.getcontext
dD = D = Decimal = decimal.Decimal
DT = decimal.DecimalTuple
dgt = decimal.getcontext()

dgt.prec = 50


def print_error (error, x=None, thisName=None) :
    &quot;&quot;&quot;
Prints error derived from sys.exc_info().
    &quot;&quot;&quot;
    list1 = error.split(',')
    v1 = ','.join(list1[1:-1])
    if thisName : print (thisName)
    if x != None : print ('  Input =', (str(x))[:60])
    print (' ', list1[0])
    print (' ', v1)
    print (' ', list1[-1])


def makeDecimal (x, flag = 0) :
    '''
output = makeDecimal (x [, flag]) 
x is a single object convertible to Decimal object.
returns Decimal object or
returns None on error.
'''
    thisName = 'makeDecimal (x) :'
    if flag :
        print ()
        print (thisName)
        print ('  Input x =', type(x), (str(x))[:60])
        
    if isinstance(x, dD) : return x
    if isinstance(x, int) : return dD(x)
    if isinstance(x, float) : return dD(str(x))
    if isinstance(x, str) :
        try :
            error = ''
            output = dD(x)
        except : error = str(sys.exc_info())
        if error :
            if flag : print_error (error) 
            return None
        return output
        
    if isinstance(x, (list,tuple,DT)) :
        try :
            error = ''
            v1,v2,v3 = x
            if isinstance(v2, (tuple,list)) : output = dD(x)
            else : output = dD( ( v1, list(v2), v3 ) )
        except : error = str(sys.exc_info())
        if error :
            if flag : print_error (error) 
            return None
        return output

    if flag :
        print ('  Input not recognized.')
    return None
&lt;/syntaxhighlight&gt;
===checkComplex(x)===

This function verifies that the object is a valid complex tuple.

&lt;syntaxhighlight lang=python&gt;
def checkComplex(x, flag = 0) :
    &quot;&quot;&quot;
status = checkComplex(x [, flag]) 
x must be :
    (v1,v2,'rectangular') or
    (v1,v2,'polar')
    v1,v2 must be type Decimal.
    &quot;&quot;&quot;
    thisName = 'checkComplex (x) :'

    if flag :
        print ()
        print (thisName)
        print ('  Input x =', type(x), (str(x))[:60])

    try:
        error = ''
        v1,v2,str1 = x
        isinstance (v1,dD) or ({}['v1 must be type Decimal.'])
        isinstance (v2,dD) or ({}['v2 must be type Decimal.'])
        (isinstance (str1,str) and str1) or ({}['str1 must be valid string.'])
        str1 = str1.lower()
        if str1 == 'rectangular'[:len(str1)] : pass
        elif str1 == 'polar'[:len(str1)] : pass
        else : ({}['str1 must be &quot;rectangular&quot; or &quot;polar&quot;.'])
    except : error = str(sys.exc_info())
    if error :
        if flag : print_error (error)
        return False
    return True
&lt;/syntaxhighlight&gt;

===str_to_complex (input_string)===

&lt;syntaxhighlight lang=python&gt;
import re

# The re pattern for numbers to be found in string of type complex.
# In string like '  1234567890987654323456E-6 + 987654321234567890987e+2J   '
# extracts '1234567890987654323456E-6' and '+987654321234567890987e+2J'.
# eval allows '  12_3456_7890_9876_5432_3456E-6 + 987654321234567890987e+2J   '.
# Conversion to Decimal allows: '__987_65432123____4567890_987___'.
# eval does not allow ' 0123 ' but allows '  2e04 '.
# Conversion to Decimal allows: ' 0123  '
digits  = '[_0123456789]'
integer = '[_0123456789]{1,}'                    # Matches 1 23 345 4567 56789
exponent = '[Ee][\+\-]{{0,1}}{}'.format(integer) # Matches e8 e+8 e-8 or E8 E+8 E-8
Exponent = '({}){{0,1}}'.format(exponent)        # exponent 0 or 1 times.
float1 = '({})\.{{0,1}}'.format(integer)         # Matches 123  2345.
float2 = '{}{{0,}}\.{}'.format(digits, integer)  # Matches   123.345  .345
float_ = '(({})|({}))'.format(float2,float1)
rvalue = '{}({})'.format(float_,Exponent)
rvalue_signed = '[\+\-]{{0,1}}{}'.format(rvalue)
ivalue_signed = '{}[Jj]'.format(rvalue_signed)


def str_to_complex (input_string) :
    &quot;&quot;&quot;
a,b = str_to_complex (input_string)
input_string could be '   1234567890987654323456E-6 + 987654321234567890987e+2J    '
input_string could be ' ' '  ( 12345678909876543234561234567890987.654323456E-6 +
                             987654321234567890987987654321.23764567890987e+2J  )  ' ' '
or ' ' '
    - (
 (
    12345678_90543123456789054_312345678905431234567890543e-014 +
    543123456789054_3123456789054312345_67890543654327453E-1_3J
) )

    ' ' '
or ' ' '
    - (
 (
    12345678_90543123456789054_312345678905431234567890543e-014 + # Note the '_'
    543123456789054_3123456789054312345_67890543654327453E-1_3J
) )

    ' ' '
or '4', '-4', '(4)', '(-4)', '-(4)', '-(-4)'
or '( +3j)', '+(-3J)', '-(3j)', '-(-3J)'
or '( 4+3j)', '+(-3 + 4J)','-( 4+3j)', '-(-3 + 4J)',
This function retains precision of a,b
    &quot;&quot;&quot;
    
    try :
        status = 0
        isinstance(input_string,str) or ({}['input_string not type str.'])
        str1 = input_string.strip()
        cx1 = eval(str1)
        isinstance(cx1,(int,float,complex)) or ({}['cx1 not desired type.'])
    except : status = 1
    if status : return

    # This code removes white lines and comments, if any, at end of each line.
    new_line = '''
'''[-1:]
    lines = [ line for Line in str1.split(new_line) for line in [ Line.rstrip() ] if line ]
    lines = [ v for line in lines
                for parts in [ line.split('#') ]
                for v in [ parts[0]] ]
    str1 = ''.join(lines)
    str1 = ''.join(str1.split())

    if isinstance (cx1, (int,float)) :
        resultr = re.search(rvalue_signed, str1)
        resultr or ({}['rvalue not found.'])
        dD1 = dD(resultr[0])
        v1 = eval(str(dD1))
        if v1 == cx1 : return dD1,dD(0)
        if v1 == -cx1 : return -dD1,dD(0)
        ({}['dD1 not recognized.'])

    # cx1 must be complex. It must contain imaginary value.
    resulti = re.search(ivalue_signed, str1)
    resulti or ({}['ivalue not found.'])
    str2j = resulti[0] ; str2 = str2j[:-1]
    dD2 = dD(str2)
    str1 = ''.join(str1.split(str2j))

    # cx1 may contain real value.
    resultr = re.search(rvalue_signed, str1)
    if resultr : dD1 = dD(resultr[0])
    else : dD1 = dD(0)

    cx2 = complex(dD1,dD2)
    if cx2.real == cx1.real : pass
    elif cx2.real == -cx1.real : dD1 = dD1.copy_negate()
    else : ({}['dD1 Not Recognized.'])
    if cx2.imag == cx1.imag : pass
    elif cx2.imag == -cx1.imag : dD2 = dD2.copy_negate()
    else : ({}['dD2 Not Recognized.'])

    cx3 = complex(dD1, dD2)
    if cx3 == cx1 : return dD1,dD2
    ({}['No match for cx3.'])

&lt;/syntaxhighlight&gt;

===makeComplex(x)===

&lt;syntaxhighlight lang=python&gt;
def makeComplex (x, flag = 0) :
    '''
result = makeComplex (x[, flag]) 

Input can be tuple with 1,2 or 3 members.
If 1 or 2 members, 'rect' is understood.
The one member or single object may be int, float, complex, CompleX
or string convertible to int, float or complex.

x = makeComplex(4)
x = makeComplex((4,))
x = makeComplex(('4',0))
x = makeComplex((4,'0', 'rect'))
x = makeComplex(4+0j)
x = makeComplex('4+0j')
x = makeComplex(('4+0j',))
In all seven cases above x = ( Decimal('4'), Decimal('0'), 'rect' )

output is always
(modulus, phase, &quot;polar&quot;) or
(real_part, imag_part, &quot;rect&quot;)
modulus, phase, real_part, imag_part are Decimal objects.

On error returns None.
'''

    thisName = 'makeComplex (x) :'

    if flag :
        print ()
        print (thisName)
        print ('  Input x =', type(x), (str(x))[:60])

    if isinstance(x, CompleX) :  # New class CompleX (note the punctuation.)
        x.check()
        return (x.r, x.i, 'rect')
    
    if isinstance (x,complex) : return makeComplex (( x.real, x.imag )) 

    try :
        status = 1
        a,b = str_to_complex (x) 
    except : status = 0
    if status : return a,b,'rect'

    try :
        status = 1
        result = makeDecimal(x)
        isinstance(result, Decimal) or ({}['Expecting result to be type Decimal.'])
        output = result,dD(0),'rect'
    except : status = 0
    if status : return output

    try :
        status = 1
        v1, = x # Allow for (( 3+4j ),)
    except : status = 0
    if status : return makeComplex (v1)

    try :
        status = 1 ; error = ''
        if len(x) == 2 :
            v1,v2 = [ makeDecimal (v) for v in x  ]
            str1 = 'rect'
        elif len(x) == 3 :
            v1,v2,str1 = x
            v1,v2 = [ makeDecimal (v) for v in (v1,v2)  ]
        else : ({}['len(x) not in (2,3)'])
        output = v1,v2,str1
        checkComplex(output) or ({}['output not valid complex.'])
        if str1[0] in 'rR' : output = v1,v2,'rect'
        else : output = v1,v2,'polar'
    except : status = 0 ; error = str(sys.exc_info())
    if status : return output
    
    if flag : print_error (error)
    return 
&lt;/syntaxhighlight&gt;

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

This section examines the parabola, ellipse and hyperbola, showing how to calculate the equation of
the section, and also how to calculate the foci and directrices given the equation.
==Latera recta et cetera==
&quot;Latus rectum&quot; is a Latin expression meaning &quot;straight side.&quot;
According to Google, the Latin plural of &quot;latus rectum&quot; is &quot;latera recta,&quot;
but English allows &quot;latus rectums&quot; or possibly &quot;lati rectums.&quot;
The title of this section is poetry to the eyes and music to the ears of a Latin student
and this author hopes that the gentle reader will permit such poetic licence in a mathematical topic.

The translation of the title is &quot;Latus rectums and other things.&quot; This section describes the calculation of interesting items 
associated with the ellipse: latus rectums, major axis, minor axis, focal chords, directrices and various points on these lines.


When given the equation of an ellipse, the first thing is to calculate eccentricity, foci and directrices as shown above.
Then verify that the curve is in fact an ellipse.

From these values everything about the ellipse may be calculated. For example:
{{RoundBoxTop|theme=2}}
[[File:0608ellipse01.png|thumb|400px|'''Graph of ellipse &lt;math&gt;1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.&lt;/math&gt;'''
&lt;/br&gt;
&lt;/br&gt;
Axis : (-0.8)x + (-0.6)y + (9.4) = 0&lt;/br&gt;
Eccentricity = 0.9&lt;/br&gt;
&lt;/br&gt;
Directrix 2 : (0.6)x + (-0.8)y + (2) = 0&lt;/br&gt;
Latus rectum RS : (0.6)x + (-0.8)y + (-0.8) = 0&lt;/br&gt;
Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0&lt;/br&gt;
Latus rectum PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0&lt;/br&gt;
Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{ID2}&lt;/math&gt; = (6.32, 7.24)&lt;/br&gt;
&lt;math&gt;\text{I2}&lt;/math&gt;  = (7.204210526315789473684, 6.061052631578947368421)&lt;/br&gt;
F2  = (8, 5)&lt;/br&gt;
M   = (15.16210526315789473684, -4.54947368421052631579)&lt;/br&gt;
F1  = (22.32421052631578947368, -14.09894736842105263158)&lt;/br&gt;
&lt;math&gt;\text{I1}&lt;/math&gt;  = (23.12, -15.16)&lt;/br&gt;
&lt;math&gt;\text{ID1}&lt;/math&gt; = (24.00421052631578947368, -16.33894736842105263158)&lt;/br&gt;
&lt;/br&gt;
P   = (20.30821052631578947368, -15.61094736842105263158)&lt;/br&gt;
Q   = (10.53708406832736953616, -8.018239580333420216299)&lt;/br&gt;
R   = (5.984, 3.488)&lt;/br&gt;
S   = (10.016, 6.512)&lt;/br&gt;
T   = (19.78712645798841993752, -1.080707788087632415281)&lt;/br&gt;
U   = (24.34021052631578947368, -12.58694736842105263158)&lt;/br&gt;
&lt;/br&gt;
Length of major axis: &lt;math&gt;\text{I1I2}&lt;/math&gt; = 26.52631578947368421052&lt;/br&gt;
Length of minor axis: QT = 11.56255298707631300170&lt;/br&gt;
Length of latus rectum: RS = PU = 5.04
]]
Consider conic section: &lt;math&gt;1771x^2 + 1204y^2 + 1944xy -44860x - 18520y + 214400 = 0.&lt;/math&gt;

This curve is ellipse with random orientation.
&lt;syntaxhighlight lang=python&gt;
# python code

ABCDEF = A,B,C,D,E,F = 1771, 1204, 1944, -44860, -18520, 214400 # ellipse

result = calculate_abc_epq(ABCDEF)
(len(result) == 2) or 1/0

# ellipse or hyperbola
(abc1,epq1), (abc2,epq2) = result
a1,b1,c1 = abc1 ; e1,p1,q1 = epq1
a2,b2,c2 = abc2 ; e2,p2,q2 = epq2
(e1 == e2) or 2/0
(1 &gt; e1 &gt; 0) or 3/0

print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) )
A,B,C,D,E,F = ABCDEF_from_abc_epq(abc1,epq1)
print ('Equation of ellipse in standard form:')
print ( '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'.format(A,B,C,D,E,F) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(1771)x^2 + (1204)y^2 + (1944)xy + (-44860)x + (-18520)y + (214400) = 0
Equation of ellipse in standard form:
(-0.7084)x^2 + (-0.4816)y^2 + (-0.7776)xy + (17.944)x + (7.408)y + (-85.76) = 0
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
# python code
def sum_zero(input) :
    '''
sum = sum_zero(input)
If sum is close to 0 and Tolerance permits, sum is returned as 0.
For example: if input contains (2, -1.999999999999999999999)
this function returns sum of these 2 values as 0.
    '''
    global Tolerance
    sump = sumn = 0
    for v in input :
        if v &gt; 0 : sump += v
        elif v &lt; 0 : sumn -= v
    sum = sump - sumn
    if abs(sum) &lt; Tolerance : return (type(Tolerance))(0)
    min, max = sorted((sumn,sump))
    if abs(sum) &lt;= Tolerance*min : return (type(Tolerance))(0)
    return sum
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
===Major axis===
&lt;syntaxhighlight lang=python&gt;
# axis is perpendicular to directrix.
ax,bx = b1,-a1
# axis contains foci. ax + by + c = 0
cx = reduce_Decimal_number(-(ax*p1 + bx*q1))
axis = ax,bx,cx
print ( '    Axis : ({})x + ({})y + ({}) = 0'.format(ax,bx,cx) )
print ( '    Eccentricity = {}'.format(e1) )
print ()
print ( '    Directrix 1 : ({})x + ({})y + ({}) = 0'.format(a1,b1,c1) )
print ( '    Directrix 2 : ({})x + ({})y + ({}) = 0'.format(a2,b2,c2) )
F1 = p1,q1 # Focus 1.
print ( '    F1 : ({}, {})'.format(p1,q1) )
F2 = p2,q2 # Focus 2.
print ( '    F2 : ({}, {})'.format(p2,q2) )

#  Direction cosines along axis from F1 towards F2:
dx,dy = a1,b1

# p2 = p1 + dx*distance_F1_F2
# q2 = q1 + dy*distance_F1_F2
if dx : distance_F1_F2 = (p2 - p1)/dx
else :  distance_F1_F2 = (q2 - q1)
if distance_F1_F2 &lt; 0 :
    distance_F1_F2 *= -1
    dx *= -1 ; dy *= -1

# Intercept on directrix1
distance_from_F1_to_ID1 = abs(a1*p1 + b1*q1 + c1)
ID1 = xID1,yID1 = p1 - dx*distance_from_F1_to_ID1, q1 - dy*distance_from_F1_to_ID1
print ( '    Intercept ID1 : ({}, {})'.format(xID1,yID1) )

# 
# distance_F1_F2
# -------------------- = e
# length_of_major_axis
# 
length_of_major_axis = distance_F1_F2 / e1

# Intercept1 on curve
distance_from_F1_to_curve = (length_of_major_axis - distance_F1_F2 )/2

xI1,yI1 = p1 - dx*distance_from_F1_to_curve, q1 - dy*distance_from_F1_to_curve
I1 = xI1,yI1 = [ reduce_Decimal_number(v) for v in (xI1,yI1) ]
print ( '    Intercept I1 : ({}, {})'.format(xI1,yI1) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Axis : (-0.8)x + (-0.6)y + (9.4) = 0
    Eccentricity = 0.9

    Directrix 1 : (0.6)x + (-0.8)y + (-27.47368421052631578947) = 0
    Directrix 2 : (0.6)x + (-0.8)y + (2) = 0
    F1 : (22.32421052631578947368, -14.09894736842105263158)
    F2 : (8, 5)
    Intercept ID1 : (24.00421052631578947368, -16.33894736842105263158)
    Intercept I1 : (23.12, -15.16)
&lt;/syntaxhighlight&gt;
{{RoundBoxTop|theme=2}}
Techniques similar to above can be used to calculate points 
&lt;math&gt;I2, ID2.&lt;/math&gt;
{{RoundBoxBottom}}

===Latus rectums===
&lt;syntaxhighlight lang=python&gt;
# direction cosines along latus rectum.
dlx,dly = -dy,dx
# 
# distance from U to F1               half_latus_rectum
# ------------------------------ = ----------------------- = e1
# distance from U to directrix 1   distance_from_F1_to_ID1
# 
half_latus_rectum = reduce_Decimal_number(e1*distance_from_F1_to_ID1)
# latus rectum 1
# Focal chord has equation (afc)x + (bfc)y + (cfc) = 0.
afc,bfc = a1,b1
cfc = reduce_Decimal_number(-(afc*p1 + bfc*q1))
print ( '    Focal chord PU : ({})x + ({})y + ({}) = 0'.format(afc,bfc,cfc) )
        
P = xP,yP = p1 + dlx*half_latus_rectum, q1 + dly*half_latus_rectum
print ( '    Point P : ({}, {})'.format(xP,yP) )
U = xU,yU = p1 - dlx*half_latus_rectum, q1 - dly*half_latus_rectum
print ( '    Point U : ({}, {})'.format(xU,yU) )
distance = reduce_Decimal_number(( (xP - xU)**2 + (yP - yU)**2 ).sqrt())
print ('    Length PU =', distance)
print ('    half_latus_rectum =', half_latus_rectum)
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Focal chord PU : (0.6)x + (-0.8)y + (-24.67368421052631578947) = 0
    Point P : (20.30821052631578947368, -15.61094736842105263158)
    Point U : (24.34021052631578947368, -12.58694736842105263158)
    Length PU = 5.04
    half_latus_rectum = 2.52
&lt;/syntaxhighlight&gt;
{{RoundBoxTop|theme=2}}
Techniques similar to above can be used to calculate points 
&lt;math&gt;R, S.&lt;/math&gt;
{{RoundBoxBottom}}

===Minor axis===
&lt;syntaxhighlight lang=python&gt;
print ()

# Mid point between F1, F2:
M = xM,yM = (p1 + p2)/2, (q1 + q2)/2
print ( '    Mid point M : ({}, {})'.format(xM,yM) )
half_major  = length_of_major_axis / 2
half_distance = distance_F1_F2 / 2

# half_distance**2 + half_minor**2 = half_major**2
half_minor = ( half_major**2 - half_distance**2 ).sqrt()
length_of_minor_axis = half_minor * 2
        
Q = xQ,yQ = xM + dlx*half_minor,  yM + dly*half_minor
T = xT,yT = xM - dlx*half_minor,  yM - dly*half_minor
print ( '    Point Q : ({}, {})'.format(xQ,yQ) )
print ( '    Point T : ({}, {})'.format(xT,yT) )

print ('    length_of_major_axis =', length_of_major_axis)
print ('    length_of_minor_axis =', length_of_minor_axis)
# 
# A basic check.
# length_of_minor_axis**2 = (length_of_major_axis**2)(1-e**2)
# 
# length_of_minor_axis**2
# ----------------------- = 1-e**2
# length_of_major_axis**2
# 
# length_of_minor_axis**2
# ----------------------- + (e**2 - 1) = 0
# length_of_major_axis**2
# 
values = (length_of_minor_axis/length_of_major_axis)**2, e1**2 - 1
sum_zero(values) and 3/0
        
aM,bM = a1,b1 # Minor axis is parallel to directrix.
cM = reduce_Decimal_number(-(aM*xM + bM*yM))
print ( '    Minor axis : ({})x + ({})y + ({}) = 0'.format(aM,bM,cM) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
    Mid point M : (15.16210526315789473684, -4.54947368421052631579)
    Point Q : (10.53708406832736953616, -8.018239580333420216299)
    Point T : (19.78712645798841993752, -1.080707788087632415281)
    length_of_major_axis = 26.52631578947368421052
    length_of_minor_axis = 11.56255298707631300170
    Minor axis : (0.6)x + (-0.8)y + (-12.73684210526315789474) = 0
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

===Checking===
{{RoundBoxTop|theme=2}}
All interesting points have been calculated without using equations of any of the relevant lines.

However, equations of relevant lines are very useful for testing, for example:


* Check that points &lt;math&gt;ID2, I2, F2, M, F1, I1, ID1&lt;/math&gt; are on axis.

* Check that points &lt;math&gt;R, F2, S&lt;/math&gt; are on latus rectum through &lt;math&gt;F2.&lt;/math&gt;

* Check that points &lt;math&gt;Q, M, T&lt;/math&gt; are on minor axis through &lt;math&gt;M.&lt;/math&gt;

* Check that points &lt;math&gt;P, F1, U&lt;/math&gt; are on latus rectum through &lt;math&gt;F1.&lt;/math&gt;


Test below checks that 8 points &lt;math&gt;I1, I2, P, Q, R, S, T, U&lt;/math&gt; are on ellipse and satisfy eccentricity &lt;math&gt;e = 0.9.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
t1 = (
    ('I1'), ('I2'),
    ('P'), ('Q'), ('R'),
    ('S'), ('T'), ('U'),
)

for name in t1 :
    value = eval(name)
    x,y = [ reduce_Decimal_number(v) for v in value ]
    print ('{} : ({}, {})'.format((name+' ')[:2], x,y))
    values = A*x**2, B*y**2, C*x*y, D*x, E*y, F
    sum_zero(values) and 3/0

    # Relative to Directrix 1 and Focus 1:
    distance_to_F1 = ( (x-p1)**2 + (y-q1)**2 ).sqrt()
    distance_to_directrix1 = a1*x + b1*y + c1
    e1 = distance_to_F1 / distance_to_directrix1
    print ('    e1 =',e1) # Raw value is printed.

    # Relative to Directrix 2 and Focus 2:
    distance_to_F2 = ( (x-p2)**2 + (y-q2)**2 ).sqrt()
    distance_to_directrix2 = a2*x + b2*y + c2
    e2 = distance_to_F2 / distance_to_directrix2
    e2 = reduce_Decimal_number(e2)
    print ('    e2 =',e2) # Clean value is printed.
&lt;/syntaxhighlight&gt;
{{RoundBoxTop|theme=2}}
Note the differences between &quot;raw&quot; values of &lt;math&gt;e_1&lt;/math&gt; and &quot;clean&quot; values of &lt;math&gt;e_2.&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
I1 : (23.12, -15.16)
    e1 = -0.9000000000000000000034 
    e2 = 0.9
I2 : (7.204210526315789473684, 6.061052631578947368421)
    e1 = -0.9 
    e2 = 0.9
P  : (20.30821052631578947368, -15.61094736842105263158) 
    e1 = -0.9 
    e2 = 0.9
Q  : (10.53708406832736953616, -8.018239580333420216299)
    e1 = -0.9000000000000000000002
    e2 = 0.9
R  : (5.984, 3.488)
    e1 = -0.9000000000000000000003
    e2 = 0.9
S  : (10.016, 6.512) 
    e1 = -0.9000000000000000000003
    e2 = 0.9   
T  : (19.78712645798841993752, -1.080707788087632415281)
    e1 = -0.8999999999999999999996
    e2 = 0.9
U  : (24.34021052631578947368, -12.58694736842105263158)
    e1 = -0.9
    e2 = 0.9
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

==Traditional definition of ellipse==
{{RoundBoxTop|theme=2}}
[[File:0617ellipse01.png|thumb|400px|'''Traditional definition of ellipse.'''
&lt;/br&gt;
Sum of distance &lt;math&gt;F_1G&lt;/math&gt; and distance &lt;math&gt;F_2G&lt;/math&gt; is constant.
]]
Ellipse may be defined as the locus of a point that moves so that the sum of its distances from two fixed points is constant.


In the diagram the two fixed points are the foci, Focus 1 or &lt;math&gt;F_1&lt;/math&gt; and Focus 2 or &lt;math&gt;F_2.&lt;/math&gt;

Distance between &lt;math&gt;F_1&lt;/math&gt; and &lt;math&gt;F_2&lt;/math&gt;, distance &lt;math&gt;F_1F_2&lt;/math&gt;, must be non-zero.

Point &lt;math&gt;G&lt;/math&gt; on perimeter of ellipse moves so that sum of distance &lt;math&gt;F_1G&lt;/math&gt;
and distance &lt;math&gt;F_2G&lt;/math&gt; is constant.


Points &lt;math&gt;T_1&lt;/math&gt; and &lt;math&gt;T_2&lt;/math&gt; are on axis of ellipse and the same rule applies to these points.

distance &lt;math&gt;F_1T_1&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt; is constant.


distance &lt;math&gt;F_1T_1&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt;

&lt;math&gt;=&lt;/math&gt; distance &lt;math&gt;F_1G&lt;/math&gt; + distance &lt;math&gt;F_2G&lt;/math&gt;

&lt;math&gt;=&lt;/math&gt; distance &lt;math&gt;F_2T_2&lt;/math&gt; + distance &lt;math&gt;T_1F_2&lt;/math&gt;

&lt;math&gt;= \text{length of major axis.}&lt;/math&gt;

Therefore the constant is &lt;math&gt;\text{length of major axis}&lt;/math&gt; which must be greater than distance &lt;math&gt;F_1F_2.&lt;/math&gt;


From information given, calculate eccentricity &lt;math&gt;e&lt;/math&gt; and equation of one directrix.
Choose directrix 1 &lt;math&gt;dx1&lt;/math&gt; associated with focus F1.


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
==Ellipse at origin==
{{RoundBoxTop|theme=2}}
[[File:0901ellipse00.png|thumb|400px|'''Traditional definition of ellipse.'''
&lt;/br&gt;
Sum of distance &lt;math&gt;F_1P&lt;/math&gt; and distance &lt;math&gt;F_2P&lt;/math&gt; is constant.
]]
Traditional definition of ellipse states that ellipse is locus of a point that moves so that sum of its distances from two fixed points is constant.


By definition distance &lt;math&gt;F_2P&lt;/math&gt; + distance &lt;math&gt;F_1P&lt;/math&gt; is constant.

&lt;math&gt;\sqrt{(x-(-p))^2 + y^2} + \sqrt{(x-p)^2 + y^2} = k\ \dots\ (1)&lt;/math&gt;

Expand &lt;math&gt;(1)&lt;/math&gt; and result is &lt;math&gt;Ax^2 + By^2 + F = 0\ \dots\ (2)&lt;/math&gt; where:

&lt;math&gt;A = 4k^2 - 16p^2&lt;/math&gt;

&lt;math&gt;B = 4k^2&lt;/math&gt;

&lt;math&gt;F = 4k^2p^2 - k^4&lt;/math&gt;


When &lt;math&gt;y = 0,&lt;/math&gt; point &lt;math&gt;B,\ Ax^2 = -F&lt;/math&gt;

&lt;math&gt;x^2 = \frac{-F}{A}&lt;/math&gt;
&lt;math&gt;= \frac{k^4 - 4k^2p^2}{4k^2 - 16p^2}&lt;/math&gt;
&lt;math&gt;=\frac{k^2(k^2-4p^2)}{4(k^2 - 4p^2)} = \frac{k^2}{4}.&lt;/math&gt;

Therefore: &lt;math&gt;x = \frac{k}{2} = a&lt;/math&gt;

&lt;math&gt;k = \text{length of major axis.}&lt;/math&gt;

By definition, distance &lt;math&gt;F_2A&lt;/math&gt; + distance &lt;math&gt;F_1A = k.&lt;/math&gt;

Therefore distance &lt;math&gt;F_1A = a.&lt;/math&gt;


Intercept form of ellipse at origin:

&lt;math&gt;(4k^2 - 16p^2)x^2 + (4k^2)y^2 = k^4 - 4k^2p^2&lt;/math&gt;

&lt;math&gt;\frac{4(k^2-4p^2)}{k^2(k^2-4p^2)}x^2 + \frac{4k^2}{k^2(k^2 - 4p^2)}y^2 = 1&lt;/math&gt;

&lt;math&gt;\frac{4}{(2a)^2}x^2 + \frac{4}{(2a)^2 - 4p^2}y^2 = 1&lt;/math&gt;

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

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

==Second definition of ellipse==
{{RoundBoxTop|theme=2}}
[[File:0901ellipse00.png|thumb|400px|'''Graph of ellipse &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt; where &lt;math&gt;a,b = 20,12&lt;/math&gt;.'''
&lt;/br&gt;
At point &lt;math&gt;B,\ \frac{u}{v} = e.&lt;/math&gt;
&lt;/br&gt;
At point &lt;math&gt;A,\ \frac{a}{t} = e.&lt;/math&gt;
]]
Ellipse is path of point that moves so that ratio of distance to
fixed point and distance to fixed line is constant.


Let &lt;math&gt;\frac{p}{a} = e&lt;/math&gt; where:

* &lt;math&gt;p&lt;/math&gt; is non-zero,

* &lt;math&gt;a &gt; p,&lt;/math&gt;

* &lt;math&gt;a = p + u.&lt;/math&gt;

Therefore, &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;

Let directrix have equation &lt;math&gt;x = t&lt;/math&gt; where &lt;math&gt;\frac{a}{t} = e.&lt;/math&gt;

At point &lt;math&gt;B:&lt;/math&gt;

&lt;math&gt;\frac{p}{p+u} = \frac{p+u}{p+u+v} = e&lt;/math&gt;

&lt;math&gt;(p+u)^2 = p(p+u+v)&lt;/math&gt;

&lt;math&gt;pp + pu + pu + uu = pp + pu + pv&lt;/math&gt;

&lt;math&gt;pu + uu = pv&lt;/math&gt;

&lt;math&gt;u(p + u) = pv&lt;/math&gt;

&lt;math&gt;\frac{u}{v} = \frac{p}{p+u} = e&lt;/math&gt;

&lt;math&gt;\frac{\text{distance to focus}}{\text{distance to directrix}} = e\ \dots\ (3)&lt;/math&gt;

Statement &lt;math&gt;(3)&lt;/math&gt; is true at point &lt;math&gt;A&lt;/math&gt; also.


Section under &quot;Proof&quot; below proves that statement (3) is true for any point &lt;math&gt;P&lt;/math&gt; on ellipse.
{{RoundBoxBottom}}
===Proof===
{{RoundBoxTop|theme=2}}
[[File:0902ellipse00.png|thumb|400px|'''Proving that &lt;math&gt;\frac{\text{distance from point to focus}}{\text{distance from point to directrix}} = e&lt;/math&gt;.'''
&lt;/br&gt;
Graph is part of curve &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1.&lt;/math&gt;
&lt;/br&gt;
distance to Directrix1 &lt;math&gt;= t - x = \frac{a}{e} - x = \frac{a - ex}{e}.&lt;/math&gt;
&lt;/br&gt;
base = &lt;math&gt;x - p = x - ae&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{(distance to Focus1)}^2 = \text{base}^2 + y^2&lt;/math&gt;
]]
As expressed above in statement &lt;math&gt;3,&lt;/math&gt; second definition of ellipse states that ellipse is path of point that moves so that ratio of distance to
fixed point and distance to fixed line is constant.


This section proves that this definition is true for any point &lt;math&gt;P&lt;/math&gt; on the ellipse.


At point &lt;math&gt;P:&lt;/math&gt;

&lt;math&gt;(a^2 - p^2)x^2 + a^2y^2 -a^2(a^2 - p^2) = 0&lt;/math&gt;

&lt;math&gt;y^2 = \frac{-(a^2 - p^2)x^2 + a^2(a^2 - p^2)}{a^2}&lt;/math&gt;

&lt;math&gt;= \frac{a^2e^2x^2 - a^2x^2 + a^2a^2 - a^2a^2e^2}{a^2}&lt;/math&gt;

&lt;math&gt;= e^2x^2 - x^2 + a^2 - a^2e^2&lt;/math&gt;

base &lt;math&gt;= x-p = x-ae&lt;/math&gt;

&lt;math&gt;(\text{distance}\ F_1P)^2 = y^2 + \text{base}^2 = y^2 + (x-ae)^2&lt;/math&gt;
&lt;math&gt;= a^2 - 2aex + e^2x^2&lt;/math&gt;
&lt;math&gt;= (a-ex)^2&lt;/math&gt;

&lt;math&gt;\text{distance to Focus1} = \text{distance}\ F_1P = a - ex&lt;/math&gt;

&lt;math&gt;\text{distance to Directrix1} = t - x = \frac{a}{e} - x = \frac{a-ex}{e}&lt;/math&gt;

&lt;math&gt;\frac{\text{distance to Focus1}}{\text{distance to Directrix1}}&lt;/math&gt;
&lt;math&gt;= (a - ex)\frac{e}{(a-ex)}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;


Similar calculations can be used to prove the case for Focus2 &lt;math&gt;(-p, 0)&lt;/math&gt; and Directrix2 &lt;math&gt;(x = -t)&lt;/math&gt;
in which case:

&lt;math&gt;\frac{\text{distance to Focus2}}{\text{distance to Directrix2}}&lt;/math&gt;
&lt;math&gt;= (a + ex)\frac{e}{(a + ex)}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;

Therefore: &lt;math&gt;\frac{\text{distance to focus}}{\text{distance to directrix}} = e&lt;/math&gt; where &lt;math&gt;1 &gt; e &gt; 0.&lt;/math&gt;


Ellipse is path of point that moves so that ratio of distance to fixed point and distance to fixed line is constant, called eccentricity &lt;math&gt;e.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

==Heading==
===Properties of ellipse===
{{RoundBoxTop|theme=2}}
[[File:0822ellipse01.png|thumb|400px|'''Graph of ellipse used to illustrate and calculate certain properties of ellipses.'''
&lt;/br&gt;
&lt;/br&gt;
Traditional definition of ellipse:
&lt;/br&gt;
&lt;math&gt;\text{distance } AF_1 + \text{distance } AF_2 = \text{constant } k.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Second definition of ellipse:
&lt;/br&gt;
&lt;math&gt;\frac{\text{distance } AF_1}  {\text{distance } AG } = \text{eccentricity } e.&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
Triangle &lt;math&gt;A F_1  G&lt;/math&gt; is right triangle.
&lt;/br&gt;
&lt;math&gt;e = \cos \angle  O F_1  A  = \cos \angle   F_1 A G&lt;/math&gt;
]]
Ellipse in diagram has:


* Two foci: &lt;math&gt;F_1\ (p,0),\ F_2\ (-p,0).&lt;/math&gt;

* Length of major axis &lt;math&gt;= \text{distance } I_2 I_1 = 2a&lt;/math&gt;

* Length of minor axis &lt;math&gt;= \text{distance } A B = 2b&lt;/math&gt;

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

* Length of latus rectum &lt;math&gt;= \text{distance } P Q&lt;/math&gt;

* Distance between directrices &lt;math&gt;= \text{distance } D_2 D_1 = 2t&lt;/math&gt;


Properties of ellipse:

* &lt;math&gt;\frac{\text{length of major axis}}  {\text{distance between directrices}} = e&lt;/math&gt;


* &lt;math&gt;\frac{\text{distance between foci}}  {\text{length of major axis}} = e&lt;/math&gt;


* &lt;math&gt;\frac{\text{distance between foci}}  {\text{distance between directrices}}= e^2&lt;/math&gt;


* &lt;math&gt;(\frac{\text{length of minor axis}}  {\text{length of major axis}})^2 + e^2 = 1&lt;/math&gt;


* &lt;math&gt;\frac{\text{length of latus rectum}}  {\text{length of major axis}} + e^2 = 1&lt;/math&gt;


* line &lt;math&gt;PD_1&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}
====Major axis====
From traditional definition of ellipse:

Distance &lt;math&gt;AF_2\ +&lt;/math&gt; distance &lt;math&gt;AF_1&lt;/math&gt;

= distance &lt;math&gt;I_1F_1\ +&lt;/math&gt; distance &lt;math&gt;I_1F_2&lt;/math&gt;

= distance &lt;math&gt;I_2F_2\ +&lt;/math&gt; distance &lt;math&gt;I_2F_1&lt;/math&gt;

= &lt;math&gt;k.&lt;/math&gt;


Therefore: 

Length of major axis = distance &lt;math&gt;I_2I_1 = 2a = k.&lt;/math&gt;

Distance &lt;math&gt;AF_1 = \frac{k}{2} = a.&lt;/math&gt;


From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }AF_1}{\text{distance }AG} = \frac{a}{t} = \text{eccentricity }e&lt;/math&gt;
&lt;math&gt;= \frac{\text{distance }OI_1}{\text{distance }OD_1}.&lt;/math&gt;

&lt;math&gt;\frac{\text{length of major axis}}{\text{distance between directrices}} = e.&lt;/math&gt;
====Foci====
From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }I_1F_1}{\text{distance }I_1D_1} = \frac{a-p}{t-a} = e.&lt;/math&gt;

&lt;math&gt;a - p = te - ae&lt;/math&gt;

&lt;math&gt;a - p = a - ae&lt;/math&gt;

Therefore: &lt;math&gt;p = ae&lt;/math&gt; or &lt;math&gt;\frac{p}{a} = e.&lt;/math&gt;

&lt;math&gt;\frac{\text{distance between foci}}{\text{length of major axis}} = e.&lt;/math&gt;

&lt;math&gt;\frac{\text{distance between foci}}{\text{distance between directrices}} = e^2.&lt;/math&gt;
====Minor axis====
Triangle &lt;math&gt;AOF_1&lt;/math&gt; is right triangle.

&lt;math&gt;\cos ^2 \angle OAF_1 + \sin ^2 \angle OAF_1&lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    (\frac{p}{a})^2  &lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    (\frac{ae}{a})^2  &lt;/math&gt;
&lt;math&gt;= (\frac{b}{a})^2    +    e^2  = 1&lt;/math&gt;


&lt;math&gt;( \frac{\text{length of minor axis}} {\text{length of major axis}} )^2 + e^2 = 1&lt;/math&gt;


Triangles &lt;math&gt;AOF_1,\ AF_1G&lt;/math&gt; are similar.

Triangle &lt;math&gt;AF_1G&lt;/math&gt; is right triangle.


&lt;math&gt;e = \cos \angle OF_1A = \cos \angle F_1AG.&lt;/math&gt;
====Latus rectum====
From second definition of ellipse:

&lt;math&gt;\frac{\text{distance }PF_1} {\text{distance }F_1D_1} = \frac{\text{distance }PF_1}{t-p} = e&lt;/math&gt;

&lt;math&gt;\text{distance }PF_1 = te - pe = a - (ae)e = a(1-e^2).&lt;/math&gt;

&lt;math&gt;\frac{\text{distance }PF_1} {a} = 1 - e^2.&lt;/math&gt;

&lt;math&gt; \frac{\text{length of latus rectum}} {\text{length of major axis}}  + e^2 = 1&lt;/math&gt;

====Slope of curve====
Curve has equation: &lt;math&gt;b^2x^2 + a^2y^2 = a^2b^2&lt;/math&gt;

Derivative of both sides:

&lt;math&gt;b^22x + a^22yy' = 0&lt;/math&gt;

&lt;math&gt;y' = \frac{-xb^2}{ya^2}&lt;/math&gt;&lt;math&gt;&lt;/math&gt;
&lt;math&gt;= \frac{-x(1-e^2)}{y}&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

At point &lt;math&gt;P:\ m_1 = y' = \frac{-p(1-e^2)}{-a(1-e^2)}&lt;/math&gt;
&lt;math&gt;= \frac{ae}{a} = e.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

Slope of line &lt;math&gt;PD_1:\ m_2 = \frac{\text{distance }PF_1}{\text{distance }F_1D_1} = e.&lt;/math&gt;&lt;math&gt;&lt;/math&gt;&lt;math&gt;&lt;/math&gt;

&lt;math&gt;m_1 = m_2.&lt;/math&gt; Therefore line &lt;math&gt;PD_1&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;

&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

===Intercept form of equation===


&lt;math&gt;\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0625ellipse01.png|thumb|400px|'''Ellipse at origin with major axis on X axis.'''
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
Equation of ellipse has format &lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1&lt;/math&gt; where:
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{Length of major axis} = 2a = \text{distance}\ I_2I_1 = 40&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{Length of minor axis} = 2b = \text{distance}\ BA = 24&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\frac{\text{Length of minor axis}}{\text{Length of major axis}} = \sqrt{1 - e^2}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;e = \sqrt{1 - \frac{b^2}{a^2}} = 0.8.&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
]]
In diagram:

Intercept &lt;math&gt;I_1&lt;/math&gt; has coordinates &lt;math&gt;(a,0).&lt;/math&gt;

Intercept &lt;math&gt;I_2&lt;/math&gt; has coordinates &lt;math&gt;(-a,0).&lt;/math&gt;

Intercept &lt;math&gt;A&lt;/math&gt; has coordinates &lt;math&gt;(0,b).&lt;/math&gt;

Intercept &lt;math&gt;B&lt;/math&gt; has coordinates &lt;math&gt;(0,-b).&lt;/math&gt;

Focus &lt;math&gt;F_1&lt;/math&gt; has coordinates &lt;math&gt;(f,0)&lt;/math&gt; where &lt;math&gt;f = ea.&lt;/math&gt;

Focus &lt;math&gt;F_2&lt;/math&gt; has coordinates &lt;math&gt;(-f,0).&lt;/math&gt;

Curve has equation &lt;math&gt;\frac{x^2}{20^2} + \frac{y^2}{12^2} = 1,&lt;/math&gt; called intercept form of equation of ellipse because intercepts are apparent as the 
fractional value of each coefficient.


Standard form of this equation is: &lt;math&gt;(-0.36)x^2 + (-1)y^2 + (0)xy + (0)x + (0)y + (144) = 0.&lt;/math&gt; While the standard form is
valuable as input to a computer program, the intercept form is still attractive to the human eye because center of ellipse and
intercepts are neatly contained within the equation.


Slope of curve:

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

Derivative of both sides:

&lt;math&gt;b^22x + a^22yy' = 0&lt;/math&gt;

&lt;math&gt;y' = \frac{-xb^2}{ya^2}&lt;/math&gt;
&lt;math&gt;= \frac{-x(1-e^2)}{y}&lt;/math&gt;


At point &lt;math&gt;P&lt;/math&gt; on latus rectum &lt;math&gt;PQ:&lt;/math&gt;

&lt;math&gt;m_1 = y' = \frac{-(ea)(1-e^2)}{-(a(1-e^2))} = e&lt;/math&gt;


Slope of line &lt;math&gt;PD = m_2 = \frac{PF_1}{F_1D} = e&lt;/math&gt;

&lt;math&gt;m_1 = m_2.&lt;/math&gt; Line &lt;math&gt;PD&lt;/math&gt; is tangent to curve at latus rectum, point &lt;math&gt;P.&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;\text{ }&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

===Example===
{{RoundBoxTop|theme=2}}
[[File:0618ellipse01.png|thumb|400px|'''Traditional definition of ellipse uses abc, epq.'''
&lt;/br&gt;
M is mid-point between F1 and F2.
&lt;/br&gt;
Point R is on minor axis.
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\frac{\text{distance from R to F1}}{\text{distance from R to directrix 1}}&lt;/math&gt;
&lt;math&gt;= e&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;= \frac{\text{half major axis}}{\text{distance from M to directrix 1}}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{distance from M to directrix 1} = \frac{\text{half major axis}}{e}&lt;/math&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;math&gt;\text{F1:}\ (1, -7)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{F2:}\ (-1.24, 0.68)&lt;/math&gt;
&lt;/br&gt;
length_of_major_axis = 10
&lt;/br&gt;
&lt;math&gt;\text{M:}\ (-0.12, -3.16)&lt;/math&gt;
&lt;/br&gt;
length_of_minor_axis = 6
&lt;/br&gt;
&lt;math&gt;\text{R:}\ (2.76, -2.32)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;e = 0.8&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{D1:}\ (1.63, -9.16)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{Directrix 1:}\ (-0.28)x + (0.96)y + (9.25) = 0&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{abc}\ =\ (-0.28,\ 0.96,\ 9.25)&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\text{epq}\ =\ (0.8,\ 1,\ -7)&lt;/math&gt;
]]
Given:
&lt;syntaxhighlight lang=python&gt;
# python code

F1 = 1, -7          # Focus 1
F2 = -1.24, 0.68    # Focus 2
length_of_major_axis = 10
&lt;/syntaxhighlight&gt;
Calculate equation of ellipse.
&lt;syntaxhighlight lang=python&gt;

F1 = p1,q1 = [ dD(str(v)) for v in F1 ]    # Focus 1
F2 = p2,q2 = [ dD(str(v)) for v in F2 ]    # Focus 2

length_of_major_axis = dD(length_of_major_axis)
half_major_axis = length_of_major_axis / 2

# Direction cosines from F1 to F2
dx = p2-p1 ; dy = q2-q1
divider = (dx**2 + dy**2).sqrt()
dx,dy = [ (v/divider) for v in (dx,dy) ]

# F2
# p2 = p1 + dx*distance_F1_F2
# q2 = q1 + dy*distance_F1_F2
distance_F1_F2 = (q2-q1)/dy
half_distance_F1_F2 = distance_F1_F2 / 2

# The mid-point
M = xM,ym = p1 + dx*half_distance_F1_F2, q1 + dy*half_distance_F1_F2

# Eccentricity:
e = distance_F1_F2 / length_of_major_axis

# distance from point R to F1                half_major_axis
# ------------------------------------ = e = -----------------------------------------
# distance from point R to Directrix 1       distance from point M to Directrix 1

distance_from_point_M_to_dx1 = half_major_axis / e

# Intersection of axis and directrix 1
D1 = xM-dx*distance_from_point_M_to_dx1,  yM-dy*distance_from_point_M_to_dx1
D1 = xD1, yD1 = [ reduce_Decimal_number(v) for v in D1 ]

# Equation of Directrix 1
# dx1 = adx1,bdx1,cdx1
adx1,bdx1 = dx, dy # Perpendicular to axis.
# adx1*x + bdx1*y + cdx1 = 0
# Directrix 1 contains point D1
cdx1 = reduce_Decimal_number( -( adx1*xD1 + bdx1*yD1 ) )

abc = adx1,bdx1,cdx1
epq = e,p1,q1
ABCDEF = ABCDEF_from_abc_epq  (abc,epq, 1)
&lt;/syntaxhighlight&gt;
Equation of ellipse in standard form:

&lt;math&gt;(-0.949824)x^2 + (-0.410176)y^2 + (-0.344064)xy + (-1.3152)x + (-2.6336)y + (4.76) = 0&lt;/math&gt;


For more insight into method of calculation and proof:
&lt;syntaxhighlight lang=python&gt;
if 1 :
    print ('F1: ({}, {})'.format(p1,q1))
    print ('F1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p1,q1))

    print ('F2: ({}, {})'.format(p2,q2))
    print ('F2: (x - ({}))^2 + (y - ({}))^2 = 1'.format(p2,q2))

    print ('length_of_major_axis =', length_of_major_axis)
    
    print ('M: ({}, {})'.format(xM,yM))
    print ('M: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xM,yM))

    # half_minor_axis**2 + half_distance_F1_F2**2 = half_major_axis**2
    half_minor_axis = (half_major_axis**2 - half_distance_F1_F2**2).sqrt()
    length_of_minor_axis = half_minor_axis * 2
    s1 = 'length_of_minor_axis' ; print (s1, '=', eval(s1))

    # Direction cosines on major axis:
    print ('dx,dy =', dx,dy)
    
    # Direction cosines on minor axis:
    dnx,dny = dy,-dx
    print ('dnx,dny =', dnx,dny)

    # One point on minor axis:
    R = xR,yR = xM + dnx*half_minor_axis, yM + dny*half_minor_axis
    print ('R: ({}, {})'.format(xR,yR))
    print ('R: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xR,yR))
    # Verify that point R is on ellipse:
    sum_zero((A*xR**2, B*yR**2, C*xR*yR, D*xR, E*yR, F)) and 1/0

    s1 = 'e' ; print (s1, '=', eval(s1))

    print ('D1: ({}, {})'.format(xD1,yD1))
    print ('D1: (x - ({}))^2 + (y - ({}))^2 = 1'.format(xD1,yD1))

    print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(adx1, bdx1, cdx1))

    print()
# For proof, reverse the process:    
    (abc1,epq1), (abc2,epq2) = calculate_abc_epq  (ABCDEF)
    a1,b1,c1 = abc1 ; e1,p1,q1 = epq1
    print ('Directrix 1: ({})x + ({})y + ({}) = 0'.format(a1, b1, c1))
    print ('Eccentricity e1: {}'.format(e1))
    print ('F1: ({}, {})'.format(p1,q1))

    print()
    a2,b2,c2 = abc2 ; e2,p2,q2 = epq2
    print ('Directrix 2: ({})x + ({})y + ({}) = 0'.format(a2, b2, c2))
    print ('Eccentricity e2: {}'.format(e2))
    print ('F2: ({}, {})'.format(p2,q2))

    print ('\nEquation of ellipse with integer coefficients:')
    A,B,C,D,E,F = [ reduce_Decimal_number(-v*1000000/64) for v in ABCDEF ]
    str1 = '({})x^2 + ({})y^2 + ({})xy + ({})x + ({})y + ({}) = 0'
    print (str1.format(A,B,C,D,E,F))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
F1: (1, -7)
F1: (x - (1))^2 + (y - (-7))^2 = 1
F2: (-1.24, 0.68)
F2: (x - (-1.24))^2 + (y - (0.68))^2 = 1
length_of_major_axis = 10
M: (-0.12, -3.16)
M: (x - (-0.12))^2 + (y - (-3.16))^2 = 1
length_of_minor_axis = 6
dx,dy = -0.28 0.96
dnx,dny = 0.96 0.28
R: (2.76, -2.32)
R: (x - (2.76))^2 + (y - (-2.32))^2 = 1
e = 0.8
D1: (1.63, -9.16)
D1: (x - (1.63))^2 + (y - (-9.16))^2 = 1
Directrix 1: (-0.28)x + (0.96)y + (9.25) = 0

Directrix 1: (0.28)x + (-0.96)y + (-9.25) = 0
Eccentricity e1: 0.8
F1: (1, -7)

Directrix 2: (0.28)x + (-0.96)y + (3.25) = 0 
Eccentricity e2: 0.8
F2: (-1.24, 0.68) 

Equation of ellipse with integer coefficients: 
&lt;/syntaxhighlight&gt;
&lt;math&gt;(14841)x^2 + (6409)y^2 + (5376)xy + (20550)x + (41150)y + (-74375) = 0&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
{{RoundBoxBottom}}

=allEqual=

{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; padding-bottom:0.2em; padding-left:0.5em; padding-right:0.75em;&quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
====Welcomee====
{{Robelbox|title=[[Wikiversity:Welcome|Welcome]]|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFF800;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;

=====Welcomen=====
{{Robelbox|title=|theme={{{theme|9}}}}}
&lt;div style=&quot;padding-top:0.25em; 
     padding-bottom:0.2em; 
     padding-left:0.5em; 
     padding-right:0.75em;
     background-color: #FFFFFF;
     &quot;&gt;
[[Wikiversity:Welcome|Wikiversity]] is a [[Wikiversity:Sister projects|Wikimedia Foundation]] project devoted to [[learning resource]]s, [[learning projects]], and [[Portal:Research|research]] for use in all [[:Category:Resources by level|levels]], types, and styles of education from pre-school to university, including professional training and informal learning. We invite [[Wikiversity:Wikiversity teachers|teachers]], [[Wikiversity:Learning goals|students]], and [[Portal:Research|researchers]] to join us in creating [[open educational resources]] and collaborative [[Wikiversity:Learning community|learning communities]]. To learn more about Wikiversity, try a [[Help:Guides|guided tour]], learn about [[Wikiversity:Adding content|adding content]], or [[Wikiversity:Introduction|start editing now]].

&lt;/div&gt;
&lt;syntaxhighlight lang=python&gt;
# python code.
if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :if a == b == c == d == e == f == g == h == 0 :
    pass
&lt;/syntaxhighlight&gt;
{{Robelbox/close}}
{{Robelbox/close}}
{{Robelbox/close}}
&lt;noinclude&gt;
[[Category: main page templates]]
&lt;/noinclude&gt;
{| class=&quot;wikitable&quot;
|-
! &lt;math&gt;x&lt;/math&gt; !! &lt;math&gt;x^2 - N&lt;/math&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;6&lt;/code&gt; || &lt;code&gt;-221&lt;/code&gt;
|-
| &lt;code&gt;&lt;/code&gt;&lt;code&gt;7&lt;/code&gt; || &lt;code&gt;-208&lt;/code&gt;
|-
|-
| &lt;code&gt;10&lt;/code&gt; || &lt;code&gt;-157&lt;/code&gt;
|-
| &lt;code&gt;11&lt;/code&gt; || &lt;code&gt;-136&lt;/code&gt;
|-
| &lt;code&gt;12&lt;/code&gt; || &lt;code&gt;-113&lt;/code&gt;
|-
| &lt;code&gt;13&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;-88&lt;/code&gt;
|-
| &lt;code&gt;26&lt;/code&gt; || &lt;code&gt;&lt;/code&gt;&lt;code&gt;419&lt;/code&gt;
|}

=Testing=
======table1======
{|style=&quot;border-left:solid 3px blue;border-right:solid 3px blue;border-top:solid 3px blue;border-bottom:solid 3px blue;&quot; align=&quot;center&quot; 
|
Hello

As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.
When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.
&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
&lt;/syntaxhighlight&gt;
|}

{{RoundBoxTop|theme=2}}
[[File:0410cubic01.png|thumb|400px|'''
Graph of cubic function with coefficient a negative.'''
&lt;/br&gt;
There is no absolute maximum or absolute minimum.
]]
Coefficient &lt;math&gt;a&lt;/math&gt; may be negative as shown in diagram.


As &lt;math&gt;abs(x)&lt;/math&gt; increases, the value of &lt;math&gt;f(x)&lt;/math&gt; is dominated by the term &lt;math&gt;-ax^3.&lt;/math&gt;

When &lt;math&gt;x&lt;/math&gt; has a very large negative value, &lt;math&gt;f(x)&lt;/math&gt; is always positive.

When &lt;math&gt;x&lt;/math&gt; has a very large positive value, &lt;math&gt;f(x)&lt;/math&gt; is always negative.


Unless stated otherwise, any reference to &quot;cubic function&quot; on this page will assume coefficient &lt;math&gt;a&lt;/math&gt; positive.
{{RoundBoxBottom}}
&lt;math&gt;x_{poi} = -1&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;
&lt;math&gt;&lt;/math&gt;

=====Various planes in 3 dimensions=====
{{RoundBoxTop|theme=2}}
&lt;gallery&gt;
File:0713x=4.png|&lt;small&gt;plane x=4.&lt;/small&gt;
File:0713y=3.png|&lt;small&gt;plane y=3.&lt;/small&gt;
File:0713z=-2.png|&lt;small&gt;plane z=-2.&lt;/small&gt;
&lt;/gallery&gt;
{{RoundBoxBottom}}


&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight lang=python&gt;
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
&lt;/syntaxhighlight&gt;

&lt;syntaxhighlight&gt;
1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
  3501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147
  0109559971605970274534596862014728517418640889198609552329230484308714321450839762603627995251407989
  6872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471
  6038689997069900481503054402779031645424782306849293691862158057846311159666871301301561856898723723
  5288509264861249497715421833420428568606014682472077143585487415565706967765372022648544701585880162
  0758474922657226002085584466521458398893944370926591800311388246468157082630100594858704003186480342
  1948972782906410450726368813137398552561173220402450912277002269411275736272804957381089675040183698
  6836845072579936472906076299694138047565482372899718032680247442062926912485905218100445984215059112
  0249441341728531478105803603371077309182869314710171111683916581726889419758716582152128229518488472
&lt;/syntaxhighlight&gt;


&lt;math&gt;\theta_1&lt;/math&gt;


{{RoundBoxTop|theme=2}}
[[File:0422xx_x_2.png|thumb|400px|'''
Figure 1: Diagram illustrating relationship between &lt;math&gt;f(x) = x^2 - x - 2&lt;/math&gt;
and &lt;math&gt;f'(x) = 2x - 1.&lt;/math&gt;'''
&lt;/br&gt;
]]
{{RoundBoxBottom}}
&lt;math&gt;O\ (0,0,0)&lt;/math&gt;

&lt;math&gt;M\ (A_1,B_1,C_1)&lt;/math&gt;

&lt;math&gt;N\ (A_2,B_2,C_2)&lt;/math&gt;

&lt;math&gt;\theta&lt;/math&gt;


&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;\begin{align}
(6) - (7),\ 4Apq + 2Bq =&amp;\ 0\\
	       2Ap + B =&amp;\ 0\\
                   2Ap =&amp;\ - B\\
\\
                     p =&amp;\ \frac{-B}{2A}\ \dots\ (8)
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;

:&lt;math&gt;\begin{align}
1.&amp;4141475869yugh\\
  &amp;2645er3423231sgdtrf\\
  &amp;dhcgfyrt45erwesd
\end{align}&lt;/math&gt;

&lt;math&gt;\ \ \ \ \ \ \ \ &lt;/math&gt;


:&lt;math&gt;
  4\sin 18^\circ
  = \sqrt{2(3 - \sqrt 5)}
  = \sqrt 5 - 1
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    <title>Motivation and emotion/Book/2017/Hardiness</title>
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      <text bytes="22955" sha1="6uda023jzfgzbz163l053vahdnvu47f" xml:space="preserve">{{title|Hardiness:&lt;br&gt;What is it and how can it help?}}
{{MECR3|1=https://www.youtube.com/watch?v=7lqjqDOX34A}}
__TOC__

== Overview ==
Hardiness is a personality trait or attribute that is important in determining the effectiveness of{{gr}} which a person can withstand stress and times of hardship (Maddi &amp; Khoshaba, 1994). Alternatively, Maddi (2007) stated that &quot;hardiness is a pattern of attitudes and skills that provides the courage and strategies to turn stressful circumstances from potential disasters into growth opportunities instead&quot;.

Hardiness has also been categorized into three separate but equally important mechanisms (Kobasa &amp; Maddi, 1982). These three mechanisms are commitment, control, and challenge.

* '''Commitment''' is the concept of self-improvement. The dedication that someone with high levels of hardiness has to finding flaws in their life, attitude or actions and their ability to actively work on improving them. Commitment can also apply to being active in other people's lives around you. The opposite to commitment is alienation, the act of not engaging or interacting with people.{{fact}}
* '''Control''' is the amount in which {{awkward}} someone allows their life experiences and past knowledge to guide them throughout current difficult situations. Someone with high levels of hardiness would see past experiences and use them to better control their outcomes of their actions and deal with stress. Control is seen as amount of responsibility and liability that one undertakes when viewing their life, choices and future. The antithesis of control is powerlessness. Powerlessness is the belief that no matter what you do something else has already determined the outcome. The concept of &quot;fate&quot; could be seen as something that someone who feels powerless could use to not take responsibility for their decisions.{{fact}}
* '''Challenge''' is the mindset that change is inevitable and that we must adapt to changing circumstances. This part of hardiness predicts that people who plan for change will be better suited when it does inevitably arrive. The opponent of challenge is security. Security is a protected and safe way of living where change is avoided and stressed about. People who require security and a routine allergic to change would have less, and develop less, hardiness.{{fact}}

These are some popular proposed factors that make up hardiness but Duckworth (2007) proposed that hardy and gritty people have four traits in common with one another; interest, practice, purpose and hope. 

* '''Interest''' means that the hardiest people have something that they are passionate about that they are able to do for pleasure alone. This could be anything that truly allows one to be happy and enjoy immensely such as a sport, art or music.

* '''Practice''' is the action of taking that interest and constantly trying to improve even if they are already at a high level. This is a commitment to oneself to work on their passion daily.

* '''Purpose''' is the realization that helping others is just as important as helping yourself. Your passion and work is important but it cannot be as important as possible without it benefiting the people around you; your family, friends, the community or even the world.

* '''Hope''' is the ability that people have to endure through hardships. When life is hard the people with hope will rise above adversity and become better people because of it, whereas people without hope have a higher chance of crumbling.

{{RoundBoxLeft}}
;Quiz
&lt;quiz display=simple&gt;
{Cooperation is one of the three Cs of hardiness proposed by Maddi (1982).
|type=&quot;[False]&quot;}
- True
+ False

{Interest, as proposed by Duckworth, means that you are interested in becoming more hardy.
|type=&quot;[False]&quot;}
- True
+ False
&lt;/quiz&gt;
{{LeftRightBoxClose}}

== What Develops Hardiness? ==
Hardiness, like most other psychological personality traits, is believed to be shaped at an early age (National Scientific Council on the Developing Child, 2010). In order to possess a higher level of hardiness childhood is where it would most effectively be developed. Southwick et al., (2016) stated that during these crucial developmental years supportive and caring parental figure(s) are key to someone having hardiness in later life. Their report goes on to say that a childhood filled with stress and little or no support can lead to heightened negative reactions to stress later in life.

== Psychological Benefits of Hardiness ==
{{expand}}

=== Social Support and Coping ===
Maddi et al., (2006) found that hardiness was positively correlated with both effective coping strategies as well as the propensity to seek out and use social support. Social support could include family and friends as well as government programs and counselors. This was mirrored by the negative relationship that hardiness was found to have with depression and anger issues{{fact}}. This was all{{huh}} found when comparing high levels of hardiness to a high level of religiousness; {{awkward}} comparing two forms of ways in which people work through struggle.

=== Academic Success ===
Hardiness has been seen to be a significant indicator of both success in performance and retention in terms of academics (Maddi, et al., 2012). This {{what}} study was done using a military college with hardiness being an important factor due to the difficult workload and physical needs of the environment (Kelly, Matthews &amp; Bartone, 2014)

=== Post-Traumatic Stress Disorder ===
Hardiness can help in preventing the development of [[w:Post Traumatic Stress Disorder|Post Traumatic Stress Disorder]] (PTSD) (Pitts et al., 2016). PTSD is a mental disorder that occurs mainly after someone experiences an extremely traumatic and terrifying event. The attributes of seeing change as a useful challenge instead of a daunting task mean that recovery and continued progress are not seen as hard and are more attainable (Bonanno, 2004 &amp; Maddi, 2006). This can be seen when lower levels of hardiness correlated positively with prominence of PTSD in the military. The opposite was found for high levels of hardiness. This means that high levels of hardiness stopped some veterans from receiving PTSD. This can also be applied to people who are not necessarily military personnel. Car accidents, witnessing crimes and being attacked can all lead to PTSD but hardiness training could negate the disorder{{fact}}.

[[File:Posttraumatic stress disorder.webm|thumb|center|A short video on Post Traumatic Stress Disorder]]

=== Depression ===
Sinha &amp; Singh (2009) recorded the [[w:Depression|depression]] levels and hardiness levels of 320 people of various ages{{where}}. The subjects were categorized into low hardiness, medium hardiness and high hardiness. As the hardiness went up the likelihood that depression would occur went down. As is common knowledge in science, correlation does not equal causation. This being said more research into this area would be interesting and could blossom into new methods of counselling for depression.
{{RoundBoxTop}}

== Summary ==
{{expand}}

==== Psychological Benefits to hardiness ====
* Social support and coping mechanisms
* Academic Success
* Mitigates the effects of Post Traumatic Stress Disorder
* Decreased likelihood of depression
{{RoundBoxBottom}}

== Physical Benefits to Psychological Hardiness ==
{{expand}}

=== Physical Health ===
Bartone, Valdes &amp; Sandvik (2016) studied 338 people involved in a security program {{where}} and assessed their physical health.  The [[w:Body Mass Index|Body Mass Index]] (BMI), cholesterol and cardiovascular condition of the participants were checked. More &quot;good&quot; cholesterol, less body fat and greater heart health were all found to be significantly linked to high levels of hardiness. That is, that the participants with high levels of hardiness showed less signs of physical deterioration and poor health{{gr}}.

This drastic difference would most likely be due to the fact that stress raises things like cortisol production in the body as well changes the way that the body processes blood, leading to health problems (Cooper &amp; Marshall, 2013).

=== Stress Mitigation ===
Roth et al., (1989) tested 373 university students {{where}}. Hardiness, along with levels of fitness were negatively and significantly correlated with levels of illness. That is, the more hardy a person is or the more that they exercised the less likely they were to become ill. The researchers theorised that hardiness in and of itself may not be enough to stop or prevent illness. However, the fact that hardiness is a good indicator of how well, or how poorly, some people are affected by life's stressors could be a contributing factor. Strong amounts of hardiness mitigate the negative things that people experience; this keeps their stress within healthy limits and lessens the physical illness associated with heightened stress.

== Prominent Studies and Theories ==
{{expand}}

=== Maddi's Personality Theory (1989) ===
People, even as early as the 1980's, were finding it harder to find meaning in life and becoming more and more secluded{{fact}}. Maddi saw these things as the result of technological, social and cultural changes in the wake of an ever-growing globalist movement. Maddi then expanded on this by proposing two main personality types in which to categorize people.

* '''Premorbid personality''' – Premorbid personality is the personality type that is most likely going to conform to social norms, have little input into their own roles and very much just do what is expected of them. They have limited imagination as well as limited understanding of who they are or meaningful self-analysis. They can be happy living this way but have a higher chance of being depressed, bored and unhappy with their life than their counterparts.{{fact}}
* '''Ideal Identity''' - The ideal identity is the opposite of the premorbid personality in that it garners a deeper understanding of itself. Someone with this personality is more creative, questions more and makes their own decisions based on what they want and less on what the people around them want from them. They are also more adaptive to changing situations and circumstances and can find more enjoyment in learning. This engagement in their own life, along with the idea that they can influence their future and are in charge of their own life, means that they are less likely to experience the negative side effects of the premorbid personality type.{{fact}}

=== Illinois Bell Telephone Company (1987) ===
Maddi (1987) studied the psychological well-being of 430 employees of the Illinois Bell Telephone Company. Around 6 years into the study the company had to let go of almost half of its staff. The researchers realized that they had an extremely fortuitous opportunity presented to them. Despite the obvious negativity of half a company being laid off there was now room for them to research the decline (or incline) of the remaining staff members in terms of mental and physical health in relation to stress.  Maddi and his team were then able to measure the stress that inevitably affected the remaining employees given their increased work obligations. Within the study group approximately two thirds reported declines in mental, physical or emotional health due to the stress of their new workload. Divorces, obesity, substance abuse and depression were significantly higher than before the lay offs occurred. This was in stark contrast to the other third of employees that reported that they had improved or maintained their health. This led Maddi to believe that the group that displayed decreased levels of heath{{sp}} lacked a high enough level of hardiness since they had not adapted to their new roles and saw them as nuisances rather than an opportunity for growth.

This is the core of hardiness. The idea that change is seen as an opportunity rather than a hindrance. Those that{{gr}} see it as simply a learning experience are able to move forward and improve, whereas those that don't can see a decline in their health{{gr}}. Maddi had proposed and theorised that low levels of hardiness would be detrimental to some {{missing}} the remaining employees. The significant rise in negative health situations made it clear that hardiness plays a very important part in our well-being and should be looked at as an area to improve.

=== Kobasa, Maddi &amp; Kahn (1982) ===
Kobasa and partners conducted one of the largest hardiness based experiments to date. 259 male business managers {{where}} were subject to a multitude of tests to determine their belief in regards to their ability to determine outcomes in life, their schedule and the severity of illness suffered within the past few years. Kobasa and the others found that the managers that perceived life events and outcomes as more malleable, a component of hardiness, were less likely to experience illness as a result of stress. 

One drawback to this study was the fact that all the participants were male. Better comparisons and deeper conclusions could have been made if the researchers had used women as well as men. This would have gone toward filling a gap that is currently around hardiness research.

=== Nordmo{{sp}} et al., (2017) ===
Nordomo et al., (2017) found that, in terms of insomnia and the symptoms of diminished sleep, hardiness is an important factor in severity. 281 sailors on a 4 month mission were measured in terms of their hardiness levels and split into two groups, high levels of hardiness and low levels of hardiness. Both during and after the mission the crews reported levels of insomnia were recorded. By the end it was made clear that there was significantly less levels of insomnia {{missing}} experienced by the group high in hardiness when compared to their less hardy counterparts. 

There have been many similar experiments into hardiness and its effects on soldiers and military personnel. Research into the mitigation of Post-Traumatic Stress Disorder symptoms using hardiness as well as lower levels of insomnia has led to many researchers to suggest that hardiness training be provided for armed forces personnel in order to protect their mental health (Dolan &amp; Adle 2006; Skomorovsky &amp; Sudom, 2011). 

{{RoundBoxTop}}
== Something to think about==
Hardiness, in regards to psychology, is a predominantly mental personality charactersitc{{sp}}. However, as seen in the above experiemnts{{sp}}, mental hardiness can in fact help with physical health and strength. Physical illnesses, stress related illness as well as things like insomnia can all be diminished significantly by psychological hardiness{{fact}}. The phenomenon of the mind being connected to the body and its processes is called psychophysiology!
{{RoundBoxBottom}}

== How to Increase Hardiness ==
Increasing ones{{gr}} hardiness has been shown through research to increase resistance to physical, emotional and psychological illness and impairment{{fact}}. In order to improve hardiness several research projects have conducted programs aimed at fostering growth in this area. Judkins &amp; Ingram (2002) and Hasel et al., (2011) conducted experiments on nurses and university students respectively. In order to improve hardiness the experimenters ran their participants through carefully planned programs. The focus of these programs was self-belief, role playing stressful situations as well as pushing forward the idea of an internal locus of control (Rotter, 1954). An Internal Locus of Control is the idea that you are in control of the situation (and more broadly, your life) and you do not have to simply allow stressful events to take place, but that you can take control and grow from the experience.

== Gender and Hardiness ==
Gender and its relationship to hardiness has been and{{sp}} area of contention in the field for quite some time (Shepperd &amp; Kashani, 1991). Although there have been quite a few studies into the area, there is no prevailing consensus about the extent of which gender predicts hardiness or how hardiness effects men when compared to women.

[[File:Talk sign.svg|thumb|right|Men and women react to stress differently]]

For example, Shepperd &amp; Kashani (1991) tested 150 teenager's levels of hardiness and their mental health. Half the study were female and the other male. The pair found that levels of stress, and in turn commitment and control, did have an effect on the health of the young men. Those boys with less stress recorded less psychological disorders or problems than their highly stressed counterparts. Men were also found to score lower in the commitment category to women but experienced less overall stress. This was an interesting finding. However, in the same study there was no relationship between stress, hardiness and health in the females of the group. This was a similar result to Caldwell, Pearson &amp; Chin (1987). They found that 'control' (Maddi, 1987) plays a part in determining the amount that men may fall ill due to stress. Strangely enough this was not replicated in the female subjects in this study either. 

== Related Factors ==
{{expand}}

=== Resilience ===
Psychological resilience is the ability to restore oneself{{sp}} to normal and move forward after change or hardship. For example, some people may take years to move past the death of a loved one whereas as some may come to terms with it within months,{{fact}}

=== Grit ===
Grit is very similar to hardiness as they are both personality traits. Hardiness is the ability to see change, adapt and use that change to learn and grow. Grit is the sheer will that some possess to keep pushing toward a goal no matter the setbacks and obstacles (Duckworth, 2007)

== Conclusion ==
Hardiness is a fascinating yet understudied personality trait. Often lumped in with similar concepts like resilience and grit, hardiness is seemingly more important than many people think. This is in regards to emotional, psychological and physical health as well as performance in areas such as academia and sport{{gr}}. This area of psychology is important in learning about the driving factors in human behavior{{vague}}. Some more study into gender hardiness differences as well as the physical benefits of high levels of hardiness would be helpful.

== References ==
{{Hanging indent|1=
Bartone, P. T., Valdes, J. J., &amp; Sandvik, A. (2016). Psychological hardiness predicts cardiovascular health. Psychology, health &amp; medicine, 21(6), 743-749.

Bonanno, G. A. (2004). Loss, trauma, and human resilience: have we underestimated the human capacity to thrive after extremely aversive events?. American psychologist, 59(1), 20.

Caldwell, R. A., Pearson, J. L., &amp; Chin, R. J. (1987). Stress-moderating effects: Social support in the context of gender and locus of control. Personality and Social Psychology Bulletin, 13(1), 5-17.

Cooper, C. L., &amp; Marshall, J. (2013). Occupational sources of stress: A review of the literature relating to coronary heart disease and mental ill health. In From Stress to Wellbeing Volume 1 (pp. 3-23). Palgrave Macmillan UK.

Dolan, C. A., &amp; Adler, A. B. (2006). Military hardiness as a buffer of psychological health on return from deployment. Military Medicine, 171(2), 93.

Duckworth, A.L.; Peterson, C.; Matthews, M.D.; Kelly, D.R. (2007). &quot;Grit: Perseverance and passion for long-term goals&quot; (PDF). Journal of Personality and Social Psychology. 92 (6): 1087–1101. 

Hasel, K. M., Abdolhoseini, A., &amp; Ganji, P. (2011). Hardiness training and perceived stress among college students. Procedia-Social and Behavioral Sciences, 30, 1354-1358.

Judkins, S. K., &amp; Ingram, M. (2002). Decreasing stress among nurse managers: A long-term solution. The Journal of Continuing Education in Nursing, 33(6), 259-264.

Kelly, D. R., Matthews, M. D., &amp; Bartone, P. T. (2014). Grit and hardiness as predictors of performance among West Point cadets. Military Psychology, 26(4), 327.

Kobasa, S. C., Maddi, S. R., &amp; Kahn, S. (1982). Hardiness and health: A prospective study. Journal of Personality and Social Psychology, 42(1), 168-177.

Maddi, S. R., &amp; Kobasa, S. C. (1984). hardy executive. Dow Jones-Irwin.

Maddi, S. R. (1987). Hardiness training at Illinois Bell Telephone. In J. P. Opatz (Ed.), Health promotion evaluation, pp. 101-1115. Stevens Point, WI: National Wellness Institute.

Maddi, S. R., Brow, M., Khoshaba, D. M., &amp; Vaitkus, M. (2006). Relationship of hardiness and religiousness to depression and anger. Consulting Psychology Journal: Practice and Research, 58(3), 148.

Maddi, S. R. (2007). Relevance of hardiness assessment and training to the military context. Military Psychology, 19(1), 61. Maddi, S. R. (2007). Relevance of hardiness assessment and training to the military context. Military Psychology, 19(1), 61.

Maddi, S. R., Matthews, M. D., Kelly, D. R., Villarreal, B., &amp; White, M. (2012). The role of hardiness and grit in predicting performance and retention of USMA cadets. Military Psychology, 24(1), 19.

National Scientific Council on the Developing Child . Early experiences can alter gene expression and affect long‐term development. Working paper no. 10, 2010. www.developingchild.harvard.edu.

Nordmo, M., Hystad, S. W., Sanden, S., &amp; Johnsen, B. H. (2017). The effect of hardiness on symptoms of insomnia during a naval mission. International Maritime Health, 68(3), 147-152.

Pitts, B. L., Safer, M. A., Russell, D. W., &amp; Castro-Chapman, P. L. (2016). Effects of hardiness and years of military service on posttraumatic stress symptoms in U.S. Army medics. Military Psychology, 28(4), 278-284.

Roth, D. L., Wiebe, D. J., Fillingim, R. B., &amp; Shay, K. A. (1989). Life events, fitness, hardiness, and health: A simultaneous analysis of proposed stress-resistance effects. Journal of Personality and Social Psychology, 57(1), 136.

Rotter, J. B. (1954). Social Learning and Clinical Psychology. Prentice-Hall.

Shepperd, J. A., &amp; Kashani, J. H. (1991). The relationship of hardiness, gender, and stress to health outcomes in adolescents. Journal of personality, 59(4), 747-768.

Sinha, V., &amp; Singh, R. N. (2009). Immunological role of hardiness on depression. Indian journal of psychological medicine, 31(1), 39.

Skomorovsky, A., &amp; Sudom, K. A. (2011). Psychological Well-Being of Canadian forces officer candidates: The unique roles of hardiness and personality. Military Medicine, 176(4), 389-396.

Southwick, S. M., Sippel, L., Krystal, J., Charney, D., Mayes, L., &amp; Pietrzak, R. (2016). Why are some individuals more resilient than others: The role of social support. World Psychiatry, 15(1), 77-79.

Roth, D. L., Wiebe, D. J., Fillingim, R. B., &amp; Shay, K. A. (1989). Life events, fitness, hardiness, and health: A simultaneous analysis of proposed stress-resistance effects. Journal of Personality and Social Psychology, 57(1), 136.
}}

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__TOC__

==Overview==
[[File:Sky diving.jpg|thumb|''Figure 1.'' Skydiving experience an example of novelty seeking. |alt=|240x240px]]

If you feel as though you prosper in conditions that might seem chaotic, or make spur of the moment decisions, like a 4 hour road trip just to get ice cream, or that you are frequently under stimulated and easily bored, chances are you fit the personality trait of a novelty seeker. 

[[wikipedia:Novelty_seeking|Novelty seeking (NS)]] is described as a temperamental trait which is associated with high impulsivity, exploratory behaviour, and is also extravagant and disorderly while also being closely related to emotionality and sensation seeking (Foulds, Boden, Newton-Howes, Mulder, &amp; Horood, 2017). 

The purpose of this chapter is to assist people in the understanding of what novelty seeking is and what motivates people to seek this out. The chapter will discuss the features of novelty seeking, the theoretical framework surrounding novelty seeking and also the positive and negative affects{{gr}} of novelty seeking. 

People may seek novelty in ways such as: 

* Skydiving as seen in Figure 1. 
* Spontaneous exploration 

While these may seem like intentional and harmless ways to seek novelty there are also ways that can be detrimental to a person's health and unintentional. These include:

* Impulsive buying for example plane tickets or booking a holiday
* quick loss of temper

'''Focus questions:''' 

# What is novelty seeking?
# What theories and research assist in understanding novelty seeking?
# How do we measure novelty seeking?
# Does the brain play a central role in novelty seeking?
# How does novelty seeking effect criminal behaviour?  

{{RoundBoxTop|theme=2}}&quot;We only know a tiny proportion about the complexity of the natural world. Wherever you look, there are still things we don't know about and don't understand. There are always new things to find out if you go looking for them&quot; David Attenborough (www.goodreads.com/quotes/tags/novelty)

{{RoundBoxBottom}}

== What is novelty seeking? ==

Novelty seeking (NS) is a personality trait mirroring excitement in response to novel stimuli with high levels of NS usually a antecedent of risk taking behaviours (Wang et al., 2015). NS is considered to be moderately heritable,whilst being situationally stable (Evern et al. 2017).  Evern and colleagues (2017) acknowledge that an individual with high NS could be described as any of the following: 

# Quick tempered 
# Excitable 
# Exploratory 
# Curious 
# Enthusiastic 
# Ardent 
# Easily bored 
# Impulsive and disorderly 

=== Measures of NS ===
There are many ways in which NS has been measured over the years. This section will delve into the checklists, models and inventory that best measure NS. 

==== Symptom Checklist-Revised (SCL-90-R) ====

* The SCL-90-R is a self report used to assess psychopathological symptoms. Consisting of 90 items rated with a 5-point likert scale, one being no problem and five being very serious to assess the extent in which individuals have experienced the listed symptoms in the last seven days

==== Cloningers{{gr}} Temperament and Character Models ====

*Cloninger proposed a psychobiological theory, including four dimensions of temperament and three dimensions of character. Initially, the model included only three temperament dimension, NS, Harm Avoidance (HA) and Reward Dependence (RD). The temperament dimensions were assumed to be independently heritable and to manifest early in development. These dimensions were defined in terms of individual differences in behavioural learning mechanisms explaining responses to novelty, danger or punishment and cues for reward (NS), avoiding aversive stimuli (HA and reactions to reward (RD). The Tri-Dimensional Personality Questionnaire (TPQ) was developed to measure this (Fruyt, Van De Wiele &amp; Van Heeringen, 2000).

==== Temperament and Character Inventory (TCI) ====

* Research into the TPQ revealed that within the RD sub-scale, persistence proved to be independent of the three temperament factors and therefore proposed as an additional fourth temperament. In order to more adequately represent individual differences, the four dimensional model was extended to a seven dimensional scheme, including three dimensions of character: Self-directedness (SD), Cooperativeness (CO) and Self-transcendence (ST). This model would be referred to as the Temperament and Character Inventory, which is a more precise and complete version of the TPQ for assessing temperament and character (Basiaux et al. 2001). 
* Within the TCI, Cloninger describes individuals high in NS as impulsive, quick tempered whenever frustrated, and prone to break the rules and regulations in order to pursue what they think will give them pleasure or thrills and those low in NS as reflective and law abiding. Descriptors of individuals who score high and low on the four temperament dimensions including NS are outlined in Table 1. Each of these explained traits, has a varying number of sub-scales. The dimensions are determined from a 240-item questionnaire (Cloninger, Svrakic &amp; Pryzbeck, 1998).

==== Temperament and Character Inventory Revised (TCI-R) ====

* This was the last psychometric instrument developed by Cloninger and colleagues, revising the TCI. In this revised version, a 5-point likert response was incorporated, and the persistence short scales were converted into a dimension with an additional new sub-scale for RD. Both versions had 240-items, but the TCI-R preserved 189 of the original TCI, 37 items were eliminated, 51 new items incorporated including five validity items ( Aluga, Blanch, Gallart &amp; Dolcet, 2010).

{| class=&quot;wikitable&quot;
|+Table 1. Descriptors of individuals who score high and low on the four temperament dimensions of the TCI
!Temperament Dimension 
!Descriptors of Extreme High
!Variants Low
|-
|Harm Avoidance
|Pessimistic 
Fearful

Shy

Fatigable 
|Optimistic
Daring

Outgoing

Energetic
|-
|Novelty Seeking
|Exploratory
Impulsive

Extravagant

Irritable
|Reserved
Rigid

Frugal

Stoical
|-
|Reward Dependence
|Sentimental
Open

Warm

Sympathetic 
|Critical
Aloof

Detached

Independent
|-
|Persistence
|Industrious
Determined

Ambitious

Perfectionist
|Lazy 
Spoiled

Underachiever

Pragmatist
|}

*

{{RoundBoxTop|2=2}} CASE STUDY: THE UGLY SIDE OF NS

NS is affected by both genetic and environmental factors, and can be measured in humans through questionnaires and in rodents using behavioural tasks. Both human and rodent studies demonstrate that high NS can predict the initiation of drug use and a transition to compulsive drug use and relapse. Wingo and colleagues look into the link between behavioural and molecular connections between novelty seeking and drug addiction (Wingo et al. 2016).

{{RoundBoxBottom|2=}} 

==Theoretical frameworks==
There are many theoretical frameworks surrounding novelty seeking. Below we delve into the main models and theories. 

===The brain ===
[[File:Four lobes animation small3.gif|alt=|thumb|''Figure 2.'' The brain plays a central role in a persons{{gr}} novelty seeking motivation.]]

* A main characteristic of discovering something new and being creative is the ability to think in ways that are different from established ways of thought (Schweizer, 2006). 
*High novelty seeking individuals and above average novelty finders are identified by a particular set of neurocognitive traits and styles of thinking (Schweizer, 2006).
* Novelty seeking behaviour is related to individual differences in specific neurotransmitter activity in the brain{{explain}}. It has also been argued that the novelty seeking personality is modulated by the transmission of the neurotransmitter dopamine (Schweizer, 2006).
*While using EEG recordings and a greyscale task{{comment|Explain - not mentioned in measurement}}, Tomer found that high NS participants showed a consistent attentional bias that favoured the right sided greyscale stimuli, while the low NS participants showed a bias to the left, suggesting hemispheric dopamine asymmetry and asymmetry in the novelty seeking behavioural approach. Participants with a higher NS personality score have shown an elevated blood oxygen dependent signal in the medical prefrontal cortex (mPFC) during emotional compared with neutral expectancy (Li et al. 2017).
*Thirty-four neurologically and psychiatrically healthy adult participants completed a study for Zald et al (2008) completing the TPQ. Magnetic resonance imaging (MRI) scans of the brain were performed alongside Positron emission tomography (PET) to assess the differences in extracellular dopamine (DA) and effects of NS in humans. Data found indicated that the NS is associated with reduced D2- like receptors availability in the substantia nigra/ventral tegmental area. Because midbrain D2-like receptors are dominated by somatodentric auto-receptors,{{gr}} these results suggest a specific inverse relationship between NS traits and auto-receptor availability. The correlation between NS personality traits and autoreceptor functioning may contribute to the increased addiction vulnerability of high NS (Zald et al. 2008). 

=== Self-determination theory ===

* Self determination theory, constructed by Deci and Ryan, is currently one of the most important motivational theories in social psychology, given considerable evidence of its capacity to predict human behaviour in multiple behavioural contexts (Gonzalez-Cutre et al. 2016)..
* A key driver of motivation set out in self-determination theory is satisfaction of three basic, psychological needs for autonomy, competence and relatedness (Gonzalez-Cutre et al. 2016).
* Gonzalez-Cutre et al. (2016) put forward the notion that the need for novelty should be added as an additional basic need alongside the needs proposed.
*[https://selfdeterminationtheory.org/wp-content/uploads/2017/04/2016_Gonzalez-Cutre_et-al_PAID.pdf See the proposed addition of novelty seeking to be added to self determination theory here.] 

[[File:Muhammad Self Determination Theory.png|center|thumb|443x443px|''Figure 3.'' Self-determination theory chart ]]

=== Contemporary approaches to the study of novelty ===

* Sensation seeking -  Zuckerman initially described as the need for varied, novel and complex sensations and experiences and the willingness to take physical and social risks for the sake of such experiences (Gonzalez-Cutre et al. 2016).
* Arnett then defines sensation seeking as the need for novelty of stimulation, giving a greater emphasis to the role of socialisation and not viewing sensation seeking as a potential for taking risks but more as a general experience present in multiple peoples lives (Gonzalez-Cutre et al. 2016)

== Quiz ==
&lt;quiz display=&quot;simple&quot;&gt;
{Which is not a descriptor of extreme high novelty seeking?
|type=&quot;()&quot;}
- Irritable
+ Open 
- Exploratory
- Impulsive

{Which of the following was not a dimension of the TPQ?
|type=&quot;()&quot;}
- Harm Avoidance
+ Self Transcendence 
- Novelty Seeking
- Reward Dependence

&lt;/quiz&gt;

==Genetic factors ==

* NS as a predisposing personality trait has been validated in animal models of multiple drugs of abuse and reflects a heritable tendency toward exploratory behaviour and desire for novel sensations (Wingo, Nesil, Choi &amp; Ming, 2015).   
* In twin and adoption studies suggest that 30-60% of variance in several personality traits is caused by inherited factors, however there is little knowledge on the number or identity of responsible genes, or how they differ between individuals or interact with developing factors to develop attitudes and actions that create a persons temperament (Benjamin et al. 1996).  
* A population association found by Ebstein et al. between a long allele of polymorphic exon III repeat sequence of the D4 dopamine receptor gene (D4DR) and the normal personality trait of novelty seeking{{explain}}.
* Possibility of a causal relationship further supported by studies undertaken by Benjamin and colleagues (Benjamin et al. 1996){{explain}}.
*In a univariate analysis of 478 heroin dependent subjects, gene variants were independently associated with both NS and age of onset of drug use: those with the TT genotype had higher NS sub-scale scores and an earlier onset age than individuals with CT or CC genotypes (Li et al. 2011){{explain}}.

== Crime and novelty seeking ==

*As seen above, Cloninger proposed models for assessing personality in a continuous tridimensional space. He hypothesised that the extreme variants of hereditary temperament traits predispose to clinical personality disorders. Cloninger also proposed that antisocial personality disorder diagnosis (ASPD) typically would feature high novelty seeking, low harm avoidance and low reward dependence. ASPD is a frequent diagnosis among the prison populations indicating that novelty seeking and crime may have a strong link (Tikkanen, Holi, Lindberg &amp; Virkkunen, 2007).
* In a study by Tikkanen and colleagues, it was aimed to test Cloninger's hypothesis and its connection to severe violence. Participants comprised of 198 male violent offenders recruited from court ordered, two month in patient mental examinations. The violent offences were generally serious, impulsive and committed under the influence of alcohol.
* Controls were 170 healthy age and gender matched volunteers recruited through newspaper advertisements, undergoing the same TPD and Diagnostic Statistical Manual evaluation as the offenders.
* Results suggested that the typical violent offender temperament profile comprised high NS, High HA and low RD.
* In line with Cloningers{{gr}} hypothesis, it was also found that trait NS was higher in ASPD than in non ASPD offenders. Similar results have emerged for substance users and psychiatric patients with cluster B disorders. 
* ASPD offenders showed particularly high impulsiveness and disorderliness, subscales of NS (Tikkanen, Holi, Lindberb &amp; Virkkunen, 2007).

=== Linking drug seeking behaviour and biological mechanisms ===

* There is considerable evidence that high novelty seekers are at increased risk for using drugs of abuse relative to low novelty seekers. A review by Bardo and colleagues examines the potential biological mechanisms that may help explain the link between NS and drug seeking behaviours (Bardo, Donohew &amp; Harrington, 1995). 
* The mesolimbic DA system is generally thought to be a critical component of the neural circuitry mediating drug reward. Evidence for the involvement of this neural system in drug reward is strongest for the psycho-stimulant and opiate drug classes, but some evidence also indicates that the mesolimbic DA system may also mediate, at least in part, the rewarding effects of sedative-hypnotics and hallucinogens(Bardo, Donohew &amp; Harrington, 1995). 
* Experimental evidence supporting a critical role of the mesolimbic DA system in mediating the rewarding effects of drugs of abuse has been derived from three major types of studies using animals  1. antagonist drug studies  2. lesioning studies   3. microdialysis studies.
* In general these studies have found that rewarding effects of psycho-stimulant drugs such as amphetamine and cocaine is most dependent upon the mesolimbic DA system, while the rewarding effect of opiate drugs appears to be partially depending upon this brain substrate (Bardo, Donohew &amp; Harrington, 1995).  
* An important point to be drawn from evolutionary theory is that NS behaviour when viewed as a phenotypic trait, varies in expression across different individuals within a given species. Individuals who are high NS may possess some advantage over low NS in locating new source of food or potential sources of danger, particularly during times of limited resources (Bardo, Donohew &amp; Harrington, 1995). 
* However it should be recognised that unbridled NS may be disadvantageous in situations where behavioural inhibition is needed to avoid predatory or dangerous events (Bardo, Donohew &amp; Harrington, 1995).
* One potential mechanism that has been offered to explain the relationship between response to inescapable novelty and response to drugs of abuse involves individual differences in the adrenocorticosterone stress axis. Corticosterone levels have been shown to modulate the effect of psycho-stimulate drugs (Bardo, Donohew &amp; Harrington, 1995). 

== Conclusion ==

#The potential that lies in new things motivates us to explore our environment for rewards and the brain learns that the stimulus, once familiar, has no reward associated with it and so it loses its potential. Only completely new objects activate the midbrain area and increase our levels of dopamine. 
# It is important to acknowledge that NS can be a very positive thing, but also can have detrimental effects. 
# NS is motivated by the environment and genetics
#Crime and NS have very strong links again, this could be due to the reward dependence, particularly drug seeking behaviour

== See also ==

* [[Motivation and emotion/Book/2014/Dopamine and motivation|Dopamine and Motivation]] (Book chapter, 2014)
* [[Motivation and emotion/Book/2010/Personality and motivation|Personality and Motivation]] (Book chapter, 2010)
* [[Motivation and emotion/Book/2016/Reward dependence and motivation|Reward Dependence and Motivation]] (Book chapter, 2016)
* [[Motivation and emotion/Book/2011/Self-determination theory|Self Determination Theory]] (Book chapter, 2011)
* [[Motivation and emotion/Book/2011/Sensation seeking|Sensation Seeking]] (Book chapter, 2011)
* [[wikipedia:Novelty_seeking|Novelty Seeking]] (Wikipedia)

== References ==

{{Hanging indent|1=

Aluja, A., Blanch, A., Gallart, S., &amp; Dolcet, J. M. (2010). The Temperament and Character Inventory Revised (TCI-R): Descriptive and factor structure in different age levels. ''Psicologia Conductual'', ''18''(2), 385

Bardo, M. T., Donohew, R. L., &amp; Harrington, N. G. (1996). Psychobiology of novelty seeking and drug seeking behaviour. ''Behavioural brain research'', ''77''(1-2), 23-43. 

Benjamin, J., Li, L., Patterson, C., Greenberg, B. D., Murphy, D. L., &amp; Hamer, D. L.   (1996). Population and familial association between the D4 dopamine receptor gene and measures of novelty seeking. ''Nature genetics, 12''(1), 81. 

Cloninger, C. R., Svrakic, D. M., &amp; Przybeck, T. R. (1998). A psychobiological model of temperament and character. ''The development of psychiatry and its complexity'', ''50''(12), 1-16

De Fruyt, F., Van De Wiele, L., &amp; Van Heering, C. (2000). Cloninger's psychobiological model of temperament and character and the five-factor model of personality. ''Personality and individual differences'', ''29''(3),441-452.

Evren, C., Alniak, I., V., Cetin, T., Umit, G., Agachanli, R., &amp; Evern, B. (2018) Relationship of probable ADHD with novelty seeking, severity of psychopathology and borderline personality disorder in a sample of patients with opioid use disorder. ''Psychiatry and Clinical Psychopharmacology'', ''28''(1), 48-55.

Foulds, J. A., Boden, J. M., Newton-Howes, G. M., Mulder, R. T., &amp; Horood, L. J.  (2017). The role of novelty seeking as a predictor of substance use disorder outcomes in early adulthood. ''Addiction,112''(9), 1629-1637. 

Gonzalez-Cutre, D., Sicilia, A., Sierra. C., Ferriz, R., &amp; Hagger, M. S. (2016) Understanding the need for novelty from a perspective of self-determination theory. ''Personality and Individual Differences, 102'', 159-169

Li, S., Demenescu, L. R., Sweeney-Reed, C. M., Krause, A. L., Metzger, C. D., &amp; Walter, M. (2017). Novelty seeking and reward dependence-related large-scale brain networks functional connectivity variation during salience expectancy. ''Human brain mapping'', ''38''(3), 4046-4077. 

Li, T., Yu, S., Du, J., Chen, H., Jiang, H., Xu, K., ... &amp; Zhao, M. (2011). Role of novelty seeking personality traits as mediator of the association between COMT and onset age of drug use in Chinese heroin dependent patients. ''PloS one'', ''6''(8), e22923

Schweizer, T. S. (2006). The psychology of novelty seeking, creativity and innovation: neurocognitive aspects within a work-psychological perspective. ''Creativity and Innovation Management, 15''(2), 164-172.

Tikkanen, R., Holi, M., Lindberg, N., &amp; Virkkunen, M. (2007). Tridimensional Personality Questionnaire data on alcoholic violent offenders: specific connections to severe impulsive cluster B personality disorders and violent criminality. ''BMC psychiatry'', ''7''(1), 36. 

Wang, Y., Liu, Y., Yang., Gu., F., Li, X., Zha, R, ... &amp; Zhang, X. (2015). Novelty seeking is related to individual risk preference and brain associated with risk prediction during decision making. ''Scientific reports, 5,'' 10534.

Wingo, T., Nesil,. T., Choi, J. S., &amp; Li, M. D. (2016). Novelty seeking and drug addiction in humans and animals: from behaviour to molecules. ''Journal of Neuroimmune Pharmacology, 11''(3), 456-470.

Zald, D. H., Cowan, R. L., Riccardi, P., Baldwin, R. M., Anasari, M. S., Li, R., .. &amp; Kessler, R. M. (2008). Midbrain dopamine receptor availability is inversely associated with novelty-seeking traits in humans. ''Journal of Neuroscience'', ''28''(53), 14372-14378. 

}}
== External links ==

* Take this quiz  [https://www.blogthings.com/areyouanoveltyseekerquiz/] to find out if you are a novelty seeker! (blogthings)
* Best selling author and renowned neuropsychiatrist Daniel Siegel explains how adolescence remodel the brain, increasing their willingness to take risks and seek out new things. Check it out here [https://www.youtube.com/watch?v=vGcFqzZYJxQ] (Youtube)
*Look into the Case Study mentioned above on the behavioural and molecular connection between [https://www.ncbi.nlm.nih.gov/pubmed/26481371 novelty seeking and drug addiction] (NCBI) 

[[Category:Motivation and emotion/Book/2019]]
[[Category:Motivation and emotion/Book/Curiosity]]
[[Category:Motivation and emotion/Book/Approach motivation]]
[[Category:Motivation and emotion/Book/Novelty]]</text>
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      <text bytes="1117" sha1="7x4cqza35jbzdzzq44dgqzej1wlqgtr" xml:space="preserve">Some of the drawbacks of the Indian education system are -

# Up to 85% of the students in India use to memorize the content given in the book and writing the same in the exams.
# Only 18% of the students use understand the concept of the subject and can able to answer in their own words.
# There is only few subjects containing practical contents.
# Not everyone has sufficient need to access the schools and higher studies.
# There are still many single teacher school in the country.
Now the changes which are needed are: 

* Indian government needs to invest large amount of money on the infrastructure of the schools.
* Methods of teaching need more changes. Teachers should encourage students to think logically and be creative.
* The biographies of successful persons should be a part of the syllabus, so that students can mould their personalities.
* Physical education should be given importance.
* There is a definite need of revolutionary changes in the Indian education system.
* With the effective learning system, India can successfully utilize its vast human resource.

[[Category:Education in India]]</text>
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      <text bytes="11748" sha1="28bmp76xn5cjlwvmnqaq884xcrezmzt" xml:space="preserve">== Also Known As ==
*Family name: Spencer-Churchill
*The family name of the [[Social Victorians/People/Marlborough | Duke of Marlborough]] is Spencer-Churchill
*This is the page for the family of Randolph Churchill and Jennie Jerome Churchill.
*Sir Winston Churchill

== Acquaintances, Friends and Enemies ==

== Timeline ==
'''1874 April 15''', Jennie Jerome and Randolph Spencer-Churchill married at the British Embassy in Paris.&lt;ref name=&quot;:0&quot;&gt;&quot;Jennie Jerome.&quot; {{Cite web|url=https://www.thepeerage.com/p10620.htm#i106192|title=Person Page|website=www.thepeerage.com|access-date=2020-11-01}}&lt;/ref&gt;

'''1895 January 29''', Randolph Spencer-Churchill died.

'''1897 July 2, Friday''', Lady Randolph Churchill attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did her sons Winston and Jack.&lt;ref name=&quot;:1&quot;&gt;Sebba, Anne. ''American Jennie: The Remarkable Life of Lady Randolph Churchill''. W. W. Norton, 2007.&lt;/ref&gt;

'''1900 June 3, Sunday, Whit Sunday''', Jennie (Lady Randolph) Churchill was present at a [[Social Victorians/Timeline/1900s#3 June 1900, Sunday|Whitsun house party at Sandringham House]]. She was &quot;just back from her hospital ship which had been a boon in South Africa, but fractiously insisting she is going to marry George Cornwallis-West.&quot;&lt;ref name=&quot;:28&quot; /&gt;{{rp|195, qting Lord Knutsford}} Leslie says, &quot;Jennie, who had been argumentative all weekend, would almost immediately marry her young George.&quot;&lt;ref name=&quot;:28&quot; /&gt;{{rp|197}}

'''1900 July 28''', Lady Randolph Churchill and George Cornwallis-West married.&lt;ref name=&quot;:0&quot; /&gt;

'''1902 August 9''', just after King Edward VII's coronation [[Social Victorians/People/Louisa Montagu Cavendish|Louise, Duchess of Devonshire]] tried &quot;to reach the Ladies' before anyone else&quot;:&lt;blockquote&gt;After the long ceremony she tried to hurry out in the wake of the royal procession, but found herself stopped by a line of Grenadier Guards. Leonie [&lt;nowiki/&gt;[[Social Victorians/People/Leslie|Leonie Leslie]]] and Jennie [Lady Randolph Churchill], who were descending from the King's special box, heard her upbraiding the officers in front of all the other peeresses, many of whom were themselves most uncomfortable. Then, trying to push her way past them, she missed her footing and fell headlong down a flight of steps to roll over on her back at the feet of the Chancellor of the Exchequer ([[Social Victorians/People/Hicks-Beach|Michael Hicks Beach]]), who stared paralyzed at this heap of velvet and ermine. The [[Social Victorians/People/de Soveral|Marquis de Soveral]] swiftly took charge of the situation and had her lifted to her feet while [[Social Victorians/People/Asquith|Margot Asquith]] nimbly retrieved the coronet, which was bouncing along the stalls, and placed it back on her head. It was a moment in which younger women naturally had to give precedence to an angry Duchess.&lt;ref name=&quot;:28&quot;&gt;Leslie, Anita. ''The Marlborough House Set''. New York: Doubleday, 1973.&lt;/ref&gt;{{rp|190}}&lt;/blockquote&gt;'''1914 April 1''', Lady Randolph Churchill and George Cornwallis-West divorced.&lt;ref name=&quot;:12&quot;&gt;{{Cite journal|date=2021-09-07|title=George Cornwallis-West|url=https://en.wikipedia.org/w/index.php?title=George_Cornwallis-West&amp;oldid=1042934380|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/George_Cornwallis-West.&lt;/ref&gt; He married Mrs. Patrick Campbell on 6 April 1914.

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==
Other members of the Spencer-Churchill family were present and are discussed on the [[Social Victorians/People/Marlborough |page for the Duke of Marlborough]].

[[File:Theodora - Basilica San Vitale (Ravenna, Italy) - croped.jpg|thumb|Detail of 6th-century mosaic icon of Theodora and attendants in the Basilica San Vitale, Ravenna, Italy]]
=== Jennie (Lady Randolph) Churchill ===
[[File:Jeanette-Jennie-Churchill-ne-Jerome-Lady-Randolph-Churchill-as-the-Empress-Theodora-wife-of-Justinian.jpg|thumb|left|alt=Black-and-white photograph of a standing woman richly dressed in an historical costume with a crown and holding an orb|Jennie, Lady Randolph Churchill as Empress Theodora, wife of Justinian. ©National Portrait Gallery, London.]]
At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], Lady Randolph Churchill was dressed as Empress Theodora of Byzantium. She was at Table 1 in the first supper seating and was in the &quot;Oriental&quot;&lt;ref&gt;“Ball at Devonshire House.” Evening ''Mail'' 05 July 1897 Monday: 8 [of 8], Col. 1a–4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003187/18970705/070/0008.&lt;/ref&gt;{{rp|p. 8, Col. 1c}} or the Duchess procession.&lt;ref&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;&lt;ref&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt;

Lafayette's portrait of &quot;Jeanette ('Jennie') Churchill (née Jerome), Lady Randolph Churchill as the Empress Theodora, wife of Justinian&quot; in costume is photogravure #193 in the [[Social Victorians/1897 Fancy Dress Ball/Photographs#The Album of Photographs|album presented to the Duchess of Devonshire]] and now in the National Portrait Gallery.&lt;ref&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.&lt;/ref&gt; The printing on the portrait says, &quot;Lady Randolph Churchill as the Empress Theodora, wife of Justinian,&quot; with a Long S in ''Empress''.&lt;ref&gt;&quot;Lady Randolph Churchill as the Empress Theodora.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158556/Jeanette-Jennie-Churchill-ne-Jerome-Lady-Randolph-Churchill-as-the-Empress-Theodora-wife-of-Justinian.&lt;/ref&gt;

The Lafayette Negative Archive has 5 poses plus some closeups of Lady Randolph in costume. They are higher resolution than the image from the album in the National Portrait Gallery but not in the public domain:

# Standing, nearly full length, masked background: http://lafayette.org.uk/chu1424.html
# Seated facing front but looking to her right: http://lafayette.org.uk/chu1468a.html
# Seated facing front but looking front, left hand raised, white flaw on the negative?: http://lafayette.org.uk/chu1468.html
# Standing, facing her right, the pose which was used for the album, but the album image appears to have a platform painted in?. Also, two closeups, one of her head and crown, the other of one of the images at the hem of her ??: http://lafayette.org.uk/chu1467b.html
# Standing, 3/4 to her left facing front, with lily in a ballet-pose hand; closeup of head: http://lafayette.org.uk/chu1467e.html

==== Descriptions of Her Costume ====
*According to the ''Carlisle Patriot'', which often has more detail than other papers, &quot;Among other Eastern Queens of ancient line was Lady Randolph Churchill as the Empress Theodora, in a dress of golden gauze thick with jewel-encrusted embroidery and wearing a high jewelled headdress, while in her right hand she carried a gold diamond-encircled ord [sic].&quot;&lt;ref&gt;&quot;Fancy Dress Ball: Unparalleled Splendour.&quot; ''Carlisle Patriot'' Friday 9 July 1897: 7 [of 8], Col. 4a–b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000365/18970709/084/0007.&lt;/ref&gt;
*&quot;Lady Randolph Churchill as the Empress Theodora, wore a diadem of quite barbaric splendour, with one large jewel resting in the middle of her forehead, and her dress was one of the great successes of the evening.&quot;&lt;ref&gt;“The Social Peepshow.” ''Gentlewoman'' 17 July 1897, Saturday: 26 [of 68], Col. 1a–b; print p. 80. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970717/145/0026.&lt;/ref&gt; (print p. 80, Col. 1a)
*Biographer Anne Sebba says she went as Empress Theodora of Byzantium: &quot;The Empress, a former courtesan as powerful as she was beautiful, was the wife of the Emperor Justinian I. She had dozens of admirers and was generally held in low regard by respectable society. Shane [Leslie, her nephew] commented somewhat cruelly that Jennie would have resembled Theodora even without fancy dress.&quot;&lt;ref name=&quot;:1&quot; /&gt;{{rp|p. 206}}

==== Commentary on Her Costume ====

=== Winston Churchill and Jack Churchill ===
Winston Churchill is pictured in the ''Gentlewoman'' story and was wearing &quot;green broché.&quot;&lt;ref name=&quot;:13&quot;&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 34, Col. 3a; 40, Col. 2b}} Jack Churchill was also present.&lt;ref name=&quot;:1&quot; /&gt; One of them was wearing a sword and fought a duel at some point that night in the garden?

== Demographics ==
*Nationality: Jennie Jerome was American, born in Brooklyn, New York&lt;ref&gt;{{Cite journal|date=2020-08-28|title=Lady Randolph Churchill|url=https://en.wikipedia.org/w/index.php?title=Lady_Randolph_Churchill&amp;oldid=975347328|journal=Wikipedia|language=en}}&lt;/ref&gt;; Randolph Spencer-Churchill was English.

== Family ==
*Jennie Jerome Spencer-Churchill, Lady Randolph Churchill (9 January 1854 – 29 June 1921)&lt;ref name=&quot;:0&quot; /&gt;
*Randolph Henry Spencer-Churchill (13 February 1849 – 24 January 1895)
#Rt. Hon. Sir Winston Leonard Spencer-Churchill (30 November 1874 – 24 January 1965)
#Major John Strange Spencer-Churchill (4 February 1880 – 23 February 1947)
*Major [[Social Victorians/People/Cornwallis-West |George Frederick Myddelton Cornwallis-West]] (14 November 1874 – 1 April 1951)&lt;ref&gt;&quot;Major George Frederick Myddelton Cornwallis-West.&quot; {{Cite web|url=https://www.thepeerage.com/p10620.htm#i106194|title=Person Page|website=www.thepeerage.com|access-date=2020-11-01}}&lt;/ref&gt;
*Montagu Phippen Porch (15 March 1877 – 8 November 1964)&lt;ref&gt;{{Cite journal|date=2026-05-25|title=Montagu Porch|url=https://en.wikipedia.org/w/index.php?title=Montagu_Porch&amp;oldid=1356027047|journal=Wikipedia|language=en}}&lt;/ref&gt;


* Sir Winston Leonard Spencer-Churchill (30 November 1874 – 24 January 1965)&lt;ref&gt;&quot;Rt. Hon. Sir Winston Leonard Spencer-Churchill.&quot; {{Cite web|url=https://www.thepeerage.com/p10620.htm#i106196|title=Person Page|website=www.thepeerage.com|access-date=2020-11-01}}&lt;/ref&gt;
* Clementine Ogilvy Hozier, Baroness Spencer-Churchill (1 April 1885 – 12 December 1977)&lt;ref&gt;&quot;Clementine Ogilvy Hozier, Baroness Spencer-Churchill.&quot; {{Cite web|url=https://www.thepeerage.com/p10620.htm#i106197|title=Person Page|website=www.thepeerage.com|access-date=2020-11-01}}&lt;/ref&gt;
*# Diana Spencer-Churchill (11 July 1909 – 19 October 1963)
*# Major Hon. Randolph Frederick Edward Spencer-Churchill (28 May 1911 – 6 June 1968)
*# Sarah Millicent Hermione Spencer-Churchill (7 October 1914 – 24 September 1982)
*# Marigold Frances Spencer-Churchill (15 November 1918 – 23 August 1921)
*# Mary Spencer-Churchill (15 September 1922 – 31 May 2014)

=== Relations ===

* Jennie Jerome Churchill was the sister of Leonie Blanche Jerome, who married [[Social Victorians/People/Leslie|Sir John Leslie]].

== Notes and Questions ==

# Lady Randolph Churchill is #132, Winston Churchill is #179 and Jack Churchill is #223 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who attended]] the Duchess of Devonshire's 2 July 1897 fancy-dress ball.

== Footnotes ==
{{reflist}}</text>
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      <text bytes="8250" sha1="j8mekqgw47s6rhtnr4o2a1usdjjc8jz" xml:space="preserve">== Overview ==
Charles Duncombe became Viscount Helmsley in 1881, when his father died and when he was 2 years old, and he did not marry until 1904. His father was not the later Earl of Feversham, and so Charles Duncombe was not the heir apparent to the earldom. His mother would still have been called Lady Helmsley or Viscountess Helmsley.

== Also Known As ==
*Family name: Duncombe
*Viscount Helmsley was a courtesy title for the eldest son and heir apparent of the [[Social Victorians/People/Feversham | Earl of Feversham]] (at the end of the 19th century).&lt;ref&gt;{{Cite journal|date=2020-10-14|title=Baron Feversham|url=https://en.wikipedia.org/w/index.php?title=Baron_Feversham&amp;oldid=983534946|journal=Wikipedia|language=en}}&lt;/ref&gt;
*Viscount Helmsley
**William Reginald Duncombe ( – 24 December 1881)
**Charles William Reginald Duncombe (24 December 1881 –1915)&lt;ref&gt;{{Cite journal|date=2020-09-12|title=Charles Duncombe, 2nd Earl of Feversham|url=https://en.wikipedia.org/w/index.php?title=Charles_Duncombe,_2nd_Earl_of_Feversham&amp;oldid=978075739|journal=Wikipedia|language=en}}&lt;/ref&gt;
*Viscountess Helmsley
**Muriel Frances Louisa Chetwynd-Talbot Duncombe (23 December 1876 – 19 January 1904
**Marjorie Blanche Eva Greville Duncombe (19 January 1904 – )
*Dowager Viscountess Helmesley
**Muriel Frances Louisa Chetwynd-Talbot Duncombe Owen (19 January 1904 – 2 March 1925)

== Acquaintances, Friends and Enemies ==
== Timeline ==
'''1876 December 23''', Muriel Frances Louisa Chetwynd-Talbot and William Reginald Duncombe married.&lt;ref name=&quot;:0&quot;&gt;&quot;Lady Muriel Frances Louisa Talbot.&quot; {{Cite web|url=https://thepeerage.com/p1278.htm#i12776|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}&lt;/ref&gt;

'''1885 June 6''', Muriel Frances Louisa Chetwynd-Talbot Duncombe and Hugh Darby Annesley Owen married.&lt;ref name=&quot;:0&quot; /&gt;

'''1896 February 12''', Mabel Theresa Duncombe and Sir William Gervase Beckett married.&lt;ref&gt;&quot;Hon. Mabel Theresa Duncombe.&quot; {{Cite web|url=https://thepeerage.com/p1727.htm#i17261|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}&lt;/ref&gt;

'''1897 July 2''', Lord and Lady Helmsley attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House. (Lord Charles Duncombe, Viscount Helmsley is #353 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who were present]]; Muriel Duncombe, Lady Helmsley is #354.)

'''1904 January 19''', Charles Duncombe and Marjorie Blanche Eva Greville married.&lt;ref name=&quot;:1&quot;&gt;&quot;Lady Marjorie Blanche Eva Greville.&quot; {{Cite web|url=https://thepeerage.com/p2289.htm#i22881|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}&lt;/ref&gt;
[[File:Martin van Meytens 003.jpg|alt=Old painting of a 9-year-old boy dressed very formally and richly, seated at a table with a crown nearby and holding a book.|thumb|Archduke Charles Joseph of Austria, c. 1747–1749.]]

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==
At the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]], Lord Charles Duncombe, Viscount Helmsley was dressed as Archduke Charles in the Archduchess Marie-Karoline and Emperor Joseph II section of the Austrian Court of Maria Theresa Quadrille.&lt;ref name=&quot;:2&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt;&lt;ref name=&quot;:3&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;

* He was in &quot;Court costume.&quot;&lt;ref&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;
* &quot;V&lt;small&gt;ISCOUNT&lt;/small&gt; H&lt;small&gt;ELMSLEY&lt;/small&gt; was in a Court costume.&quot;&lt;ref&gt;“Additional Costumes Worn at the Duchess of Devonshire’s Fancy Ball.” The ''Queen, The Lady’s Newspaper''17 July 1897, Saturday: 63 [of 97 BNA; p. 138 on the print page], Col. 2a–3a [3 of 3 cols.]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18970717/283/0064.&lt;/ref&gt;{{rp|Col. 3a}}

Muriel Duncombe, Lady Helmsley was dressed as Princess Charlotte of Lorraine, also in the Austrian Court of Maria Theresa Quadrille.&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; Muriel Duncombe was not Charles's wife but his mother.

No photographs of their costumes exist at this time.

=== Who They Were Dressed As ===
If Charles Duncombe, Viscount Helmsley was dressed as Archduke Charles Joseph of Austria, second son of Maria Therese and Francis I, then Marie-Karoline and Emperor Joseph II were his historical siblings. Charles Duncombe was 18 at the time of the ball, and Archduke Charles Joseph was nearly 16 when he died. He is shown at perhaps 9 years old in a portrait by Martin van Meytens (right).

Who Muriel, Lady Helmsley was dressed as is more difficult to determine. An Anne Charlotte of Lorraine-Brionne, known as Mademoiselle de Brionne, was at the court of Marie Antoinette and would more likely have been in the Countess of Warwick's procession. A few other Princess Charlottes or Princess Anne Charlottes of Lorraine existed. They were all at least one generation older but associated with the French rather than Austrian court. So it is not clear who she was dressed as.

== Demographics ==
*Nationality: English

== Family ==

* William Ernest Duncombe, 1st Earl Feversham of Ryedale (28 January 1829 – 13 January 1915)&lt;ref&gt;{{Cite web|url=https://www.thepeerage.com/p1873.htm#i18721|title=&quot;William Ernest Duncombe, 1st Earl Feversham of Ryedale.&quot; Person Page 1872|website=www.thepeerage.com|access-date=2026-06-09}}&lt;/ref&gt;
* Mabel Violet Graham (15 February 1833 – 28 August 1915)
*# Lady Ulrica Duncombe ( – 27 April 1935)
*# '''William Reginald Duncombe, Viscount Helmsley''' (1 August 1852 – 24 December 1881)
*# Hon. James Henry Duncombe (20 October 1853 – 10 January 1886)
*# Hon. Hubert Ernest Valentine Duncombe (14 February 1862 – 21 October 1918)
*# Lady Hermione Wilhelmina Duncombe (30 March 1864 – 19 March 1895)
*# Lady Helen Venetia Duncombe (1866 – 16 May 1954)
*# Lady Mabel Cynthia Duncombe (1869 – 25 April 1926)


*Muriel Frances Louisa Chetwynd-Talbot Duncombe Owen (c. 1860 – 2 March 1925)&lt;ref name=&quot;:0&quot; /&gt;
*William Reginald Duncombe, Viscount Helmsley (1 August 1852 – 24 December 1881)&lt;ref&gt;{{Cite journal|date=2019-08-11|title=William Duncombe, Viscount Helmsley|url=https://en.wikipedia.org/w/index.php?title=William_Duncombe,_Viscount_Helmsley&amp;oldid=910373349|journal=Wikipedia|language=en}}&lt;/ref&gt;
*#'''Mabel Theresa Duncombe''' (1877–1913)
*#'''Charles William Reginald Duncombe''', 2nd [[Social Victorians/People/Feversham | Earl of Feversham]] (1879–1916)
*Hugh Darby Annesley Owen ( – 12 March 1908)&lt;ref&gt;&quot;Hugh Darby Annesley Owen.&quot; {{Cite web|url=https://thepeerage.com/p1873.htm#i18722|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}&lt;/ref&gt;


*Charles William Reginald Duncombe, 2nd Earl of Feversham (8 May 1879 – 15 September 1916)&lt;ref&gt;&quot;Charles William Reginald Duncombe, 2nd Earl of Feversham of Ryedale.&quot; {{Cite web|url=https://thepeerage.com/p2288.htm#i22880|title=Person Page|website=thepeerage.com|access-date=2020-11-24}}&lt;/ref&gt;
*Marjorie Blanche Eva Greville Duncombe (25 October 1884 – 25 July 1964)&lt;ref name=&quot;:1&quot; /&gt;
#Lady Mary Diana Duncombe (19 March 1905 – October 1943)
#Charles William Slingsby Duncombe, 3rd Earl of Feversham (2 November 1906 – 4 September 1963)
#Hon. David William Ernest Duncombe (8 February 1910 – September 1927)

== Notes and Questions ==
#Muriel Duncombe Owen may have been Viscountess Helmsley, or Lady Helmsley. Her son Charles William Reginald Duncombe was Viscount Helmsley by this time, but he did not marry until 1904, so no other Lady Helmsely seems likely. She married Hugh Darby Annesley Owen, however, in 1885, but if she retained her title, then she would still be eligible to use it.

== Footnotes ==
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    <title>Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts</title>
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      <text bytes="149210" sha1="sfrzft7uz6k830epm4req736mltobzj" xml:space="preserve">=The Processions and Quadrilles=
After they arrived and had been greeted by the 20-year old [[Social Victorians/People/William Angus Drogo Montagu|William, Duke of Manchester]] (grandson of [[Social Victorians/People/Louisa Montagu Cavendish|Louisa, Duchess of Devonshire]]) at the bottom of the famous Devonshire House stairs and the Duke and Duchess of Devonshire at the top, the Royals led the processions into &quot;the White and Gold Saloon,&quot; where a dais had been set up for them.&lt;ref name=&quot;:0&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4A–8 Col. 2B. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;{{rp|p. 7, Col. 4c?}} 

Once the Royals were on the dais, the processions began, perhaps followed by or perhaps including the quadrilles. The ''Gentlewoman'' emphasized the &quot;Oriental&quot; procession, which was the first to be presented (the newspapers used the word ''Oriental'' to refer to what we would now think of as the Middle East and northern Africa as well as what we would now call ''Asian''): &lt;blockquote&gt;First came the Oriental queens, headed by the Duchess of Devonshire herself, who was accompanied by the Duke, as Charles V. of Germany, in black velvet and furs. Among the most magnificent of the Oriental personages was Princess Henry Pless, who, as the Queen of Sheba, was gorgeous to behold. Her dress was of purple and gold-shot gauze, bodice and skirt embroidered nearly to the knees, the train being one mass of jewels encrusted in gold. An Assyrian headdress, with clusters of diamonds over each ear, jewelled feathers, and chains of diamonds and turquoises, which were attached to armlets from shoulder to wrist, completed a costume of dazzling splendour. The other Queen of Sheba, who was Lady Cynthia Graham, was charmingly attired in white and silver and rose red. There were also two Cleopatras — Lady de Grey was one mass of beautiful embroideries, and Mrs. Arthur Paget looked her character to the life, and her jewels were quite the most magnificent in the room. Mr. Gerald Paget walked beside her, attired very effectively as Mark Antony. Among the gods and goddesses was Titania, the Queen of the Fairies; Lady Westmorland who made the prettiest Hebe; the Furies, Lady Lurgan and Lady Sophie Scott; and Lady Archibald Campbell, who elected to appear as Diana.&lt;br /&gt;&lt;br /&gt;Then came the processions of the various Courts, who afterwards formed into separate quadrilles.&lt;ref name=&quot;:8&quot;&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' Saturday 10 July 1897: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032. Print pp. 48–58.&lt;/ref&gt;{{rp|p. 32, Col. 2c}}&lt;/blockquote&gt;

The reports in the ''Times'' and the ''Gentlewoman'' agree that first the processions were presented to the Prince and Princess of Wales, and then the quadrilles were danced in front of the royals.&lt;ref name=&quot;:5&quot; /&gt;{{rp|p. 12, Col. 1a}} Dancing quadrilles was a custom at other fancy-dress balls or costume parties as well. (One type of quadrille is the American square dance.) Not everyone was part of a procession, but the quadrilles, some of which had been rehearsed at least to some degree, seem usually to have been smaller groups of people.

== The Courts ==
The processions were made up of the members of the &quot;Courts&quot; of the various monarchs, particularly queens, as well as other groups not led by the 4 queens identified by the ''Times''. The first procession was the &quot;Oriental&quot; one, which included [[Social Victorians/People/Louisa Montagu Cavendish|Louisa, Duchess of Devonshire]] as Zenobia, the Queens of Sheba and the Cleopatras. This procession was followed by the goddesses and gods.

Contrasting this ball with the fancy-dress ball hosted by the Prince and Princess of Wales at Marlborough House on 22 July 1874, the ''Times'' says,&lt;blockquote&gt;the innovation of yesterday was the idea of different Courts headed by various well-known ladies and attended by their friends as Princes and courtiers. The Royal party itself fell in very readily with this idea, and attended in historical and mostly Royal costumes of the 16th century. There were four Courts strictly so-called, besides two groups which were separately arranged, but which are only to be called Courts by an extension of the term. The four were the Elizabethan Court, headed by Lady Tweedmouth as Queen Elizabeth with Sir Francis Jeune as Lord Chief Justice, Lord Arran a Cardinal, and [Col. 1a / Col. 1b] Lord Rowton as Archibishop Farrer; the Louis XV. and XVI. Court, with Lady Curzon as Queen Marie Leczinska and Lady Warwick as Marie Antoinette; the Court of Maria Theresa with Lady Londonderry as the Empress, Lord Lansdowne as Prince Kaunitz, and Lady Lansdowne as Lady Keith; and the Court of the Empress Catherine II of Russia, its Imperial centre being Lady Raincliffe. Of equal importance with these Courts were the group of Orientals and the Italian procession, the chief members of the former being the hostess herself, the Duchess of Devonshire as Zenobia, Lady de Grey as Lysistrate, and Lady Cynthia Graham as the Queen of Sheba; while the latter, which covered not only the great period of Italian art but the 17th century as well, was made illustrious both by the beauty of the dresses and by the great distinction of many of those who wore them.&lt;ref name=&quot;:5&quot; /&gt;{{rp|p. 12, Col. 1a–2b}}&lt;/blockquote&gt;

Referencing the article on the ball in the ''Times'', both the ''Westminster Gazette'' and the London Evening ''Mail'' say, &lt;blockquote&gt;THE VARIOUS &quot;COURTS.&quot;

It is twenty-three years (the ''Times'' continues) since a ball of similar design and magnificence was given. We are referring to the famous ball at Marlborough House on July 22, 1874. In one respect there was a considerable difference, for, whereas the Prince of Wales's ball had a number of distinct quadrilles — a Venetian quadrille, a Vandyck quadrille, and a pack-of-cards quadrille — the innovation of yesterday was the idea of different Courts headed by various well-known ladies and attended by their friends as princes and courtiers. ...

The dancing was of the most desultory description. In the quadrilles people did their best to vie with the old-fashioned courtliness and grace. Some of their courtesies were quite beautifully done. In the procession everyone saluted the Princess of Wales in appropriate style. The Orientals spread out their hands in the impressive Oriental manner; the gods struck the ground with their sticks, and so on.&lt;ref&gt;“The Duchess’s Costume Ball.” ''Westminster Gazette'' 03 July 1897 Saturday: 5 [of 8], Cols. 1a–3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002947/18970703/035/0005.&lt;/ref&gt;{{rp|p. 5, Col. 2–3}} &lt;ref name=&quot;:9&quot;&gt;“Ball at Devonshire House.” Evening ''Mail'' 05 July 1897 Monday: 8 [of 8], Col. 1a–4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003187/18970705/070/0008.&lt;/ref&gt;{{rp|p. 8, Col. 1a–b}}
&lt;/blockquote&gt;A report in ''Truth'' emphasizes the queens among the costumed guests. Framed as a letter to &quot;Amy&quot; and referring obliquely to the extensive newspaper coverage of the ball, the report begins,&lt;blockquote&gt;DEAREST AMY, — The historic and fancy ball at Devonshire House outshone, as the moon the stars, every other social event of the week. I must try to describe some of the dresses for you, and am sending a sheaf of newspapers from which you will gather some idea of the splendour of the occasion. In tissue of silver and cloth of gold, and richly jewelled from head to foot, stood the stately Zenobia, Duchess of Devonshire, at the head of her marble stairway, to receive her guests of all the ages: queens who had stepped out of history to grace the scene, queens from the idyllic stories of the long ago, queens from ancient Persia and Abyssinia, and queens from Fairyland. Was not Titania there herself, with glittering wings and lily-wand? And the beautiful fair-haired queen, before whom all bent and performed obeisance as she passed, fair Marguerite de Valois, in gleaming snowy satin and high lace collar, with silver-lined train of cloth of gold, was she not our own Princess, the Queen of Hearts?&lt;ref name=&quot;:10&quot; /&gt;&lt;/blockquote&gt;The Princess of Wales was &quot;our own Princess, the Queen of Hearts.&quot; Titania &quot;with glittering wings and lily-wand&quot; may have been one of three women:

# [[Social Victorians/People/Arthur Stanley Wilson#Jack and Susanna Wilson Graham Menzies|Susannah Wilson Graham Menzies]], whose costume included &quot;an immense spray of white lilies&quot;&lt;ref name=&quot;:1&quot; /&gt;{{rp|p. 5, Col. 7b}} as a kind of very large wand or staff; her costume does not, however, seem to have wings.
# [[Social Victorians/People/Murray#Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball|Mary Graham Murray]]: neither wings nor wand is mentioned in the scant coverage in the press of her costume.&lt;ref name=&quot;:8&quot; /&gt;
# [[Social Victorians/People/de Alcalo Galiano#Mademoiselle de Alcalo Galiano|Mademoiselle de Alcalo Galiano]], whose portrait is in the album, although no newspaper descriptions of her costume exist at this time. What Mademoiselle de Alcalo Galiano is wearing on her shoulders could be interpreted as glittering wings, perhaps, but no wand or lilies are present in the portrait, and because her photographer was Bassano, other poses or images of her in costume do not seem to exist at this time. 
The ''Western Gazette'' describes the quadrilles and processions in the introduction of its story on the ball:&lt;blockquote&gt;The most sumptuous epochs of the most sumptuous Courts were represented, and that with a dazzling completeness which made the times live again all their glitter of precious stuffs, of gold brocades, and imposing arrays of jewels beyond price. There was a noble diversity, yet a satisfying consistency, for the main theory was to reproduce the Courts of princes famous for their love of sumptuary display, and notably the reign of Queen Elizabeth, as perhaps the richest in brave apparel, with the contemporaneous outlook of the French and Spanish Courts of the same era, and these afforded every opportunity for gorgeous display. It might be said that as a panorama of historical costume on these lines no such opportunity has ever occurred of seeing and realising the glories of dress, and the consistent reproduction of historical personages in all their traditional bravery to the fullest advantage.&lt;ref name=&quot;:2&quot;&gt;&quot;The Duchess of Devonshire's Great Ball. Remarkable Social Function. Crowds of Mimic Kings &amp; Queens. Panorama of Historical Costume. An Array of Priceless Jewels.&quot; ''Western Gazette'' 9 July 1897: 2 [of 8], Col. 7A–C. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000407/18970709/009/0002.&lt;/ref&gt;{{rp|p. 2, Col. 7a}}&lt;/blockquote&gt;

=== How the Courts Were Organized ===
The suggestion that some of the women form courts, which may have come from the Duchess of Devonshire herself,&lt;ref&gt;Wilson, Verity. ''Dressing Up: A History of Fancy Dress in Britain''. Reaktion, 2022: 62.&lt;/ref&gt; caused the ball to be visually organized in a way that it would not have been otherwise, because so many of the costumes were from the same time periods. Dressing as a queen was not only not unusual, but, at the many fancy-dress balls and Gothic revival tournaments, “One of the most common costumes for a lady of the Victorian period was … that of a Queen.”&lt;ref&gt;Thrush, Nanette. &quot;Clio's Dressmakers: Women and the Uses of Historical Costume.&quot; In Meaghan Clarke, ed. ''Fashionability, Exhibition Culture and Gender Politics: Fair Women''. Routledge, 2020: 258-277.&lt;/ref&gt;{{rp|270}} There were actual royals present — the Prince of Wales and his family as well as expatriate royals living in London and dignitaries from the Empire. Put on the dais and the object of formal presentations by the processions and quadrilles, [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#The Royals On the Dias|the actual royals]] mostly did not assert their royalty fictionally. The highest fictional rank among the royals was Alexandra, Princess of Wales, who was dressed as Marguerite de Valois.

The total number of women dressed as queens is large but, according to the ''Times'', only 4 defined the courts, &quot;strictly so-called.&quot;&lt;ref name=&quot;:5&quot; /&gt;{{rp|p. 12, Col. 1a–2b}} The first 4 — Marie Thérèse of Austria, Catherine II of Russia, Marie Antoinette and Queen Elizabeth — were the most important and organized in the ball, and their &quot;courts&quot; accounted for many of the other guests who attended.

Of the 700 or so people who attended the ball, 134–137 are accounted for by the first 4 queens and 263–267 by all the various kinds of courts, processions and quadrilles. (The numbers of people in the various courts are not perfectly stable: not all the newspapers that treat the courts agree on who was in them; these numbers are based on the typeset visualizations in the ''Morning Post''.)

Just because of chance and the individual choice of whom to personate, many of the others at the ball who came as individuals or part of much smaller groups and who were not in processions would have contributed to the number of people dressed in that time period and looking as if they could have been part of the courts. A large number of individuals, including almost all the royals ('''check Princess Louise, Faust costume, opera?'''), were also in Elizabethan dress. The ''Western Gazette'' says that &quot;the French and Spanish Courts [were] of the same era.&quot;&lt;ref name=&quot;:2&quot; /&gt;{{rp|p. 2, Col. 7a}} These additional costumes from the time periods of the major courts probably made the ball look more coherent, although one newspaper account describes the effect of random and unrelated people seen side by side in conversation ('''find this'''). Also, individuals dressed as ancestors represent belonging to a kinship group rather than a social network of friends.

Nearly fifty women came as historical, Biblical, and occasionally fictional queens, empresses and other regents. They were

# [[Social Victorians/People/Louisa Montagu Cavendish|Louise, Duchess of Devonshire]], dressed as Zenobia, Queen of Palmyra and functioning as an individual, maybe but not probably part of the &quot;Oriental&quot; procession
# [[Social Victorians/People/Tweedmouth|Fanny Marjoribanks, Lady Tweedmouth]], dressed as Elizabeth, Queen of England and leading 40 people
# Theresa Vane-Tempest-Stewart, [[Social Victorians/People/Londonderry|Marchioness of Londonderry]], dressed as Marie Thérèse of Austria, Queen and Holy Roman Empress and leading a procession of between 34 and 37 people
# Grace Denison, [[Social Victorians/People/Londesborough|Viscountess Raincliffe]], dressed as Catherine II, Queen of Russia and leading the Russia procession of 31 people plus trumpeters and &quot;Black Attendants&quot;
# [[Social Victorians/People/Warwick|Daisy, Countess Warwick]], dressed as Marie Antoinette, Queen of France and leading 29 Louis XV and XVI royals and courtiers, not counting pages
# [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]], dressed as Marguerite de Valois and with a court of 7 or 8 people, all family members, plus 2 attendants
# [[Social Victorians/People/Mar and Kellie|Violet, Countess of Mar and Kellie]], dressed as Beatrice and leading the Venetians procession of 47 people
# [[Social Victorians/People/Gerard|Lady Mary Gerard]], dressed as Astarte, Goddess of the Moon, not exactly a queen but in this list because she led the 7-person procession of goddesses from mythology
# [[Social Victorians/People/Zetland|Lilian, Marchioness of Zetland]], dressed as Henrietta Maria, Queen of Charles I of England and leading 12 people in the courts of Charles I and Charles II
# Elizabeth [[Social Victorians/People/Ormonde|Butler]], the [[Social Victorians/People/Ormonde|Marchioness of Ormonde]], dressed as Queen Guinevere and leading 21 people in the Knights of the Table of King Arthur procession
#[[Social Victorians/People/Rodney|Corisande Evelyn Vere, Lady Rodney]], dressed as Queen Guinevere, was not listed as being in the procession; she attended with her husband, who was dressed as King Arthur.
#Daisy Cornwallis-West, [[Social Victorians/People/Pless|Princess Henry of Pless]], dressed as Queen of Sheba and leading a procession of “Oriental” queens (23 people) with Lady Cynthia Graham
#[[Social Victorians/People/Feversham|Lady Cynthia Graham]], dressed as Queen of Sheba and leading a procession of “Oriental” queens (23 people) with Daisy, Princess Henry of Pless
#Katherine Osborne, [[Social Victorians/People/Leeds|Duchess of Leeds]], not a queen but in this list because she led the 17-person procession of Duchesses with Georgina, Dowager Countess of Dudley
#Georgina, [[Social Victorians/People/Dudley|Dowager Countess of Dudley]], not a queen but in this list because she led the 17-person procession of Duchesses with Katherine, Duchess of Leeds
# [[Social Victorians/People/Paget Family|Lady Minnie Paget]], dressed as Cleopatra in the &quot;Oriental&quot; procession; attended with her husband and her brother-in-law, who was dressed as Marc Anthony
# [[Social Victorians/People/Gwladys Robinson|Gwladys Robinson]], Marchioness of Ripon (when Countess de Grey), dressed as Cleopatra
#[[Social Victorians/People/Newcastle|Kathleen Pelham-Clinton, Duchess of Newcastle]], dressed as Princess Dashkova
# [[Social Victorians/People/Connaught|Princess Louise, Duchess of Connaught]], dressed as Anne of Austria, Queen of France
# [[Social Victorians/People/Francis Duke of Teck|Princess Mary Adelaide, Duchess of Teck]], dressed as Princess Sophia Electress of Luneberg and Hanover, mother of George I
# Madame [[Social Victorians/People/Baudon de Mony|Baudon de Mony]], dressed as Princess of Navarre
# [[Social Victorians/People/Santurce|Jesusa Murrieta del Campo Mello y Urritio, Marquisa de Santurce]], dressed as the Infanta of Spain
# [[Social Victorians/People/Clary Aldringen|Thérèse née Kinsky, Countess Clary-Aldringen]], dressed as the Queen of Naples, Napoleon's sister
# [[Social Victorians/People/Kinsky|Princess of Löwenstein-Wertheim-Rosenberg]], née Countess Josephine Kinsky, dressed as Princess Pauline Borghese, Napoleon's sister
# [[Social Victorians/People/Fitzgerald|Amelia, Lady Fitzgerald]], dressed as Marie Joséphe, Queen of Poland, A.D. 1737
# [[Social Victorians/People/Jersey|Margaret Child-Villiers, Countess of Jersey]], dressed as Anne of Austria, Queen of France
# [[Social Victorians/People/Katharine Mary Montagu Douglas Scott|Katharine Montagu-Douglas-Scott]], dressed as Marie Stuart, Mary Queen of Scots
# [[Social Victorians/People/Minto|Mary, Countess of Minto]], dressed as '''Princess Andrillon'''
# [[Social Victorians/People/Muriel Wilson|Muriel Wilson]], dressed as Queen Vashti
# [[Social Victorians/People/Tweeddale|Candida Louise, Marchioness of Tweeddale]], dressed as Empress Josephine
# Aileen, Countess of Meath, dressed as Queen Hortense
# [[Social Victorians/People/Buckingham and Chandos|Alice, Duchess of Buckingham and Chandos]], dressed as Caterina Cornaro, Queen of Cyprus
# [[Social Victorians/People/Lathom|Alice, Countess of Lathom]], dressed as Catherine of Aragon
# [[Social Victorians/People/Leslie|Leonie Blanche Jerome, Lady Leslie]], dressed as Brunhild
# Helena, Countess of Stradbroke, dressed as Delilah
# Ethel, Lady Knaresborough, dressed as Elizabeth, Queen of Bohemia
# [[Social Victorians/People/Churchill|Jennie, Lady Randolph Churchill]], dressed as Empress Theodora, wife of Justinian
# [[Social Victorians/People/Essex|Adela, Countess of Essex]], dressed as Berenice, Queen of Palestine
# Hon. Julia Beatrice Maguire, dressed as Dido, Queen of Carthage
# [[Social Victorians/People/Bischoffsheim|Clarisse Bischoffsheim]], dressed as Anne of Austria, queen with Louis XIII of France
# [[Social Victorians/People/de Trafford|Violet, Lady de Trafford]], dressed as Semiramis, Queen of Assyria
# [[Social Victorians/People/Hartopp|Millicent, Lady Cradock-Hartopp]], dressed as the Empress Josephine
# [[Social Victorians/People/Cadogan|Beatrix, Countess Cadogan]], dressed as Elizabeth, Queen of Bohemia
# [[Social Victorians/People/Somerset|Susan Margaret, Duchess of Somerset]], dressed as Jane, Queen of England, wife to King Henry VIII and mother of King Edward VI
# [[Social Victorians/People/Ampthill|Emily Theresa, Lady Ampthill]], dressed as the Princess de Lamballe
# [[Social Victorians/People/Gordon-Lennox|Blanche, Lady Gordon-Lennox]], dressed as the Princess de Lamballe
# [[Social Victorians/People/Dudley|Rachel, Countess of Dudley]], dressed as Queen Esther
# Mademoiselle de Alealo Galiano, dressed as the Queen of the Fairies
# Susannah Graham Menzies, dressed as Titania, Queen of the Fairies

Some women came as goddesses:

# [[Social Victorians/People/de Courcel|Marie, Baroness de Courcel]], dressed as Night
# Florence, Lady Terence Blackwood, dressed as Flora Goddess of Flowers
# Probably (Sybil Aimée) Geraldine Webber (née Magniac), dressed as Dawn
# [[Social Victorians/People/Fanny Ronalds|Fanny Ronalds]], dressed as Euterpe
# [[Social Victorians/People/Borthwick|Alice, Lady Glenesk]], dressed as Egeria
# [[Social Victorians/People/Gosford|Louisa Augusta Beatrice (née Montagu), Countess of Gosford]], dressed as Minerva (period of Louis XV)
# [[Social Victorians/People/Westmorland|Sybil Mary (née St Clair-Erskine), Countess of Westmorland]], dressed as Hebe
# [[Social Victorians/People/Gerard|Mary Emmeline Laura (née Milner), Lady Gerard]], dressed as Astarte, Goddess of the Moon
# [[Social Victorians/People/Hope-Vere|Marie Elizabeth Françoise Hope-Vere]] (née Guillemin), dressed as Medusa
# [[Social Victorians/People/Herschell|Agnes Adela (née Kindersley), Lady Herschell]], dressed as Night
# Dorothy Blanche ('Doreen', née Boyle), Viscountess Long, dressed as Urania, Goddess of Astronomy
# [[Social Victorians/People/Wolverton|Edith Amelia (née Ward), Lady Wolverton]], dressed as Britannia
# [[Social Victorians/People/Stanley#Lord Stanley and Lady A. Stanley|Lady Alice Stanley]], dressed as Diana

Men also came as kings and emperors:

# [[Social Victorians/People/Spencer Compton Cavendish|Spencer Compton, Duke of Devonshire]], dressed as the Emperor Charles V 
# [[Social Victorians/People/Boulatzell|N. Boulatzell]], dressed as Prince of Mingrelia 
# [[Social Victorians/People/Wolverton|Frederic Glyn, 4th Baron Wolverton]], dressed as King Richard Coeur de Lion
# [[Social Victorians/People/Duleep Singh|Prince Victor Albert Jay Duleep Singh]], dressed as Akbar
# [[Social Victorians/People/Londonderry#Castlereagh|Charles Stewart Henry Vane-Tempest-Stewart, 7th Marquess of Londonderry]] when Viscount Castlereagh as the Emperor Francis Joseph of Austria, dressed as 
# [[Social Victorians/People/Rodney|George Rodney, 7th Baron Rodney]], dressed as King Arthur of the Round Table
# [[Social Victorians/People/Dunville|John Dunville]], dressed as the Emperor Yuan of China
# [[Social Victorians/People/Reuben David Sassoon|Reuben David Sassoon]], dressed as a Persian Prince
# [[Social Victorians/People/Crewe-Milnes|Robert Offley Ashburton Crewe-Milnes, 1st Marquess of Crewe]], dressed as Philip II of Spain
# Sir Ralph Barrett Macnaghten, 9th Bt., dressed as Jerome Buonaparte, King of Westphalia
# [[Social Victorians/People/Rothschild Family|Alfred Charles de Rothschild]], dressed as King Henry III
# [[Social Victorians/People/Cavendish Bentinck|Lord Henry Cavendish-Bentinck]], dressed as the King of Poland
# [[Social Victorians/People/Hartopp|Sir Charles Edward Cradock-Hartopp, 5th Bt.]], dressed as Napoleon I
The courts or groupings are subnetworks within the network at this ball: there are other women dressed as goddesses, for example, than the ones included in the procession, suggesting that the ones who organized into groups did it based on relationships with each other than with a preference for a particular time or person.

At this ball, women were “arbiters” of cultural, social and political power. Even though both the Duke and Duchess of Devonshire hosted the ball, it was and has since always been called her ball. As social organizers, they were the gatekeepers to the aristocracy, granting some admittance and denying others.&lt;ref name=&quot;:11&quot; /&gt; As accomplished beauties and leaders of fashion, they were cultural arbiters.&lt;ref&gt;Clarke, Meaghan. ''Fashionability, Exhibition Culture and Gender Politics: Fair Women''. Routledge, 2020.&lt;/ref&gt; Costumed as queens, they presented themselves as “makers of history.&quot;&lt;ref&gt;Felber, Lynette, ed. ''Clio's Daughters: British Women Making History, 1790-1899''. Associated University Presses, 2007.&lt;/ref&gt; Their portraits taken in costume, which can be found in the National Portrait Gallery today, were a performance of wealth and privilege, and, with the identities personated, power.

=== The Courts in Performance ===
We know almost nothing about how these processions or quadrilles were formed, except that the Duchess of Devonshire may have been encouraged these women to form courts: 

# Theresa Vane-Tempest-Stewart, [[Social Victorians/People/Londonderry|Marchioness of Londonderry]] as Marie Thérèse of Austria
# Grace Denison, [[Social Victorians/People/Londesborough|Viscountess Raincliffe]], dressed as Catherine II, Queen of Russia
# [[Social Victorians/People/Warwick|Daisy, Countess Warwick]], dressed as Marie Antoinette, Queen of France
# [[Social Victorians/People/Tweedmouth|Fanny Marjoribanks, Lady Tweedmouth]] as Elizabeth, Queen of England

We know that the groups doing quadrilles would have been expected to have rehearsed, and we know that the Elizabethan procession, at least, did do so at a dinner party the night before the ball. 
== The Quadrilles ==

&quot;[T]he quadrilles took place&quot; after or as part of the procession.&lt;ref name=&quot;:0&quot; /&gt;{{rp|p. 7, Col. 4C}} A quadrille is a choreographed &quot;square&quot; dance. (The very specific kind of dance called a square dance in the U.S. is a quadrille, but not all quadrilles are American square dances.) Typically, quadrilles were made up of four couples. Apparently fancy-dress balls often included quadrilles, especially those with costumes of the past and ''bals poudres'' (typically balls with 18th-century costumes and powdered hair). The ''Western Gazette'' describes the quadrilles under &quot;The Dancing&quot;:&lt;blockquote&gt;The dancing was of the most desultory description. There was the Royal quadrille and there were the quadrilles danced by the Venetians, and the Russian quadrille. In the quadrilles the dancers did their best to vie with the old-fashioned courtliness and grace. Some of their courtesies were quite beautifully done. If all did not fall in with the spirit of their times it was excusable, as there were no rehearsals. Not until the chaperons and those who merely went to see had left or gone down to supper was there space for ordinary dancing, and even then so few of the dresses were fitted for the waltz and the gardens were so temptingly cool with all their coloured lights that the latter attracted the majority.&lt;ref name=&quot;:2&quot; /&gt;{{rp|p. 2, Col. 7c}}&lt;/blockquote&gt;

This article suggests that there was a &quot;Royal quadrille,&quot; suggesting that the Royals danced at some point as a group, and that the people did not rehearse their quadrilles.

== The Royals On the Dias ==
The Royals who were on the dais were likely the  immediate family of the Prince of Wales, including his siblings and children as well as the Princess of Wales, but a number of people who were — or had been — royals in other countries were also present at the ball.

=== The Prince and Princess of Wales's Children and Their Families ===
*[[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] as Grand Master of the Knights Hospitalier of Malta in the court of Queen Elizabeth
*[[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]] as Margaret of Valois
*Prince George of Wales, [[Social Victorians/People/George and Mary#George, Duke of York|Duke of York]] as George Clifford, 3rd Earl of Cumberland and thus in Elizabethan dress
*Mary of Teck, [[Social Victorians/People/George and Mary#Mary, Duchess of York|Duchess of York]] as a lady in attendance on Margaret of Valois (Alexandra, Princess of Wales)
*Princess Louise, [[Social Victorians/People/Fife|Duchess of Fife]] as one of the ladies of the court of Margaret of Valois (Alexandra, Princess of Wales)
*Alexander Duff, [[Social Victorians/People/Fife|Duke of Fife]] as a courtier of late Elizabethan Period, the time of Henri II
*[[Social Victorians/People/Prince Charles of Denmark|Maud of Wales, Princess Charles of Denmark]] accompanying the Princess of Wales as one of the ladies of the court of Margaret of Valois
*[[Social Victorians/People/Prince Charles of Denmark|Prince Charles of Denmark]] accompanying the Prince and Princess of Wales as a gentleman of the Court of Denmark in the time of Elizabeth
*[[Social Victorians/People/Princess Victoria of Wales|Princess Victoria of Wales]]

=== The Prince of Wales's Siblings and Their Families ===
*[[Social Victorians/People/Christian of Schleswig-Holstein|Princess Helena, or Princess Christian of Schleswig-Holstein]] as Sophia Charlotte, daughter of the Electress Sophia of Hanover and sister of George I
*[[Social Victorians/People/Christian of Schleswig-Holstein|Prince Christian of Schleswig-Holstein]] as the Earl of Lincoln in the time of Elizabeth
*[[Social Victorians/People/Victoria of Schleswig-Holstein|Princess Victoria of Schleswig-Holstein]], daughter of Helena and Christian of Schleswig-Holstein, as a lady or princess of the Elizabethan Court
*[[Social Victorians/People/Princess Louise|Princess Louise]], Marchioness of Lorne as a character from the opera ''Faust'' or the Tudor period
*John Campbell, [[Social Victorians/People/Argyll|Marquis of Lorne]] as a Tudor
*[[Social Victorians/People/Connaught|Prince Arthur, Duke of Connaught and Strathearn]] as an Elizabethan military commander
*[[Social Victorians/People/Connaught|Princess Louise, Duchess of Connaught]] as Ann of Austria
*[[Social Victorians/People/Alfred of Edinburgh|Alfred, Hereditary Prince of Saxe-Coburg and Gotha]], nephew of the Prince of Wales and son of Alfred of Edinburgh and Grand Duchess Maria Alexandrovna Romanova, as Duke Robert of Normandy, A.D. 1060
Many of these costumes are from the Elizabethan period, but the royals wearing them would not have been in the Elizabethan procession or quadrille. One newspaper report noticed this as well.

=== Other Royals Possibly on the Dais ===
These people were closely related but not of Victoria's immediate family and thus perhaps not eligible for the same obeisances? So perhaps they were not on the dais. Also, they are not listed as having marched in a procession or danced in a quadrille.

* [[Social Victorians/People/Francis Duke of Teck|Francis, Duke of Teck]]  as Capitaine Garde du Roi, 1660
* Princess [[Social Victorians/People/Francis Duke of Teck|Mary Adelaide, Duchess of Teck]] as Princess Sophia, Electress of Luneburg and Hanover
* Prince [[Social Victorians/People/Francis Duke of Teck|Alexander of Teck]] as a Dragoon Guard with a blue coat, Queen Anne's period
* Prince [[Social Victorians/People/Francis Duke of Teck|Francis of Teck]] as a Dragoon Guard with a red coat, Queen Anne's period
* Prince [[Social Victorians/People/Francis Duke of Teck|Adolphus of Teck]]
==Processions==
Speaking of the processions before the [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]] and [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]] the ''Morning Post'' describes the scene:&lt;blockquote&gt;One after the other they entered by one door, advanced up the middle of the ball room, made obeisance, and left by another door. Those who did not belong to any particular group lined the room and crowded the doorways. After this the quadrilles took place.&lt;ref name=&quot;:0&quot; /&gt;{{rp|7, Col. 4C}}&lt;/blockquote&gt;The ''Graphic'' published an illustration by W. Hatherell and J. Gulich showing a member of one of the processions bowing before the Prince and Princess of Wales, who is inclining her head in return. The guest has two train-bearers who look like children, which if this illustration is true to life means that the woman bowing is probably  

(&quot;Duchess of Devonshire's Costume Ball, The: The Procession of Guests Bowing to the Royal Group in the White and Gold Saloon. Drawn by W. Hatherell and J. Gulich.&quot; The Graphic 10 July 1897: 17–18 [of 34]. [https://www.britishnewspaperarchive.co.uk/viewer/BL/0000057/18970710/021/0017?browse=true https://www.britishnewspaperarchive.co.uk/viewer/BL/0000057/18970710/021/0017].)

It seems that the processions came in to the White and Gold Saloon and then formed in front of the Royals to do their quadrilles, so any group might be called a procession or a quadrille, depending on what exactly they did. According to the ''Morning Post'', which in this list apparently mixes up &quot;courts&quot; and processions, &quot;The following processions were formed shortly after the assembling of the guests, and passed through the ball-room&quot;:
# &quot;Oriental&quot; (their word for it, and repeated twice more later in the list)
# Goddesses and gods
#Duchess
# Venetians
# Austrian
# Russian, led by the &quot;Trumpeters of the Imperial Guard&quot;
# Louis XVI
# Elizabethan

The ''Morning Post'' story highlighted the &quot;Oriental&quot; procession, which was the first procession to dance before the Prince and Princess of Wales. The story in the ''Gentlewoman'' also emphasizes this procession by listing it first and describing it in detail. Also, the ''Morning Post'' article attempted to illustrate how people were arranged in the processions by the way their names were typeset. The ''Times'' listed people in the various processions and courts, but did not attempt a typeset visualization the way the ''Morning Post'' did.

=== &quot;Oriental&quot; Procession ===
What the newspapers called the &quot;Oriental&quot; procession was &quot;the Oriental Queens of an era previous to Christianity, with their suites,&quot; who were permitted to assemble in a different place than everybody else before leading the rest of the processions and quadrilles.&lt;ref&gt;“The Duchess’s Costume Ball.” ''Westminster Gazette'' 03 July 1897 Saturday: 5 [of 8], Cols. 1a–3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002947/18970703/035/0005.&lt;/ref&gt;{{rp|5, Col. 2c}} &lt;ref name=&quot;:8&quot; /&gt;{{rp|32, Col. 2a}} 

One newspaper, the ''Times'', says that [[Social Victorians/People/Louisa Montagu Cavendish|Louisa, Duchess]] and [[Social Victorians/People/Spencer Compton Cavendish|Spencer Cavendish, Duke of Devonshire]] took part in the &quot;Oriental&quot; group.&lt;ref name=&quot;:5&quot; /&gt;{{rp|Col. 1b}} The ''Gentlewoman'' says,&lt;blockquote&gt; 

First [to present themselves to the Royals] came the Oriental queens, headed by the Duchess of Devonshire herself, who was accompanied by the Duke, as Charles V. of Germany, in black velvet and furs.&lt;ref name=&quot;:8&quot; /&gt;{{rp|32, Col. 2c}}&lt;/blockquote&gt; 

This sentence from the ''Daily Telegraph'' occurs at the end of the description of the Duchess of Devonshire's costume, suggesting but not saying that the &quot;Oriental&quot; queens got this special treatment because she was one of them:&lt;blockquote&gt;Masters of the Ceremonies in Louis Seize military uniforms passed the guests through into inner rooms, only the Oriental Queens of an era previous to Christianity, with their suites, assembling in the white and gold saloon, with its fine pictures in the panels, and brilliantly-lighted by hundreds of wax candles in crystal chandeliers, as were all the rooms.&lt;ref&gt;“Historic Ball at Devonshire House. Brilliant Scene.” The ''Daily Telegraph'' 3 July 1897, Saturday: 9 [of 14], Col. 6a–7c [of 7]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/18970703/094/0009.&lt;/ref&gt;{{rp|9, Col. 6a}}&lt;/blockquote&gt;

The people in the &quot;Oriental Procession&quot; assembled in the white-and-gold saloon.&lt;ref name=&quot;:8&quot; /&gt;{{rp|32, Col. 2a}} The presentations to the Royals took place in the saloon or ballroom, which was &quot;particularly magnificent, being a splendid harmony of white and gold.&quot;&lt;ref name=&quot;:8&quot; /&gt;{{rp|32, Col. 2a}} Besides having the white-and-gold saloon in which to assemble, the queens and suites of the &quot;Oriental Procession&quot; lined the &quot;centre of the saloon&quot; for the procession of the Prince and Princess of Wales to their place on the dais. According to the ''Gentlewoman'',
&lt;blockquote&gt;About half-past eleven the Blue Hungarian Band, which was stationed in a small ante-room, announced the Prince of Wales' arrival with the stirring strains of &quot;God Save the Queen,&quot; and His Royal Highness led the Princess up the centre of the saloon, which was lined by ladies dressed as Oriental queens and their suites.&lt;ref name=&quot;:8&quot; /&gt;{{rp|32, Col. 2a}}&lt;/blockquote&gt;

Other than the account in the ''Gentlewoman'', most newspapers that mention it say the &quot;Oriental&quot; procession was led by Lady Cynthia Graham and [[Social Victorians/People/Pless|Daisy Cornwallis-West, Princess of Pless]], as Queens of Sheba.&lt;ref name=&quot;:5&quot; /&gt; &lt;ref name=&quot;:0&quot; /&gt;{{rp|7, Col. 5b}} In these reports, Lady Cynthia Graham's name is listed first in spite of the Princess of Pless's higher rank. ''Truth'' says that 3 women came dressed as Queen of Sheba but names only Lady Cynthia Graham and Daisy Cornwallis-West, the same 2 named by the ''Morning Post'' and ''Times'': &quot;There were three Queens of Sheba, and Paris himself could scarcely have decided to which the apple of beauty should have been awarded.&quot;&lt;ref name=&quot;:10&quot; /&gt;{{rp|42, Col. 1b}} 

==== The Attendants for the Queens of the &quot;Oriental&quot; Procession ====
Some of the Duchess of Devonshire's attendants and hired staff for the ball were people of color, probably boys and men. Some of the people in attendance on some of the queens at the ball were people of color, again probably boys and men. While not all of these queens were in the &quot;Oriental&quot; procession, several were.

[[Social Victorians/People/Pless#Attendants of Daisy, Princess Henry of Pless|Daisy, Princess of Pless had attendants in her suite]], some of whom were probably hired as well as her sister and brother, [[Social Victorians/People/Cornwallis-West|Shelagh Cornwallis-West]], as her &quot;Ethiopian attendant,&quot;&lt;ref name=&quot;:1&quot; /&gt;{{rp|p. 5, Col. 7c}} and [[Social Victorians/People/Cornwallis-West#George Cornwallis-West|George Cornwallis-West]], who was ashamed of his blackface and costume and is not mentioned in any newspaper report.

Both the ''Morning Post'' and the ''Times'' list the Hon. George Keppel in the suite of men, with the other men at his level, but the ''Gentlewoman'' says that at least 2 of the men — Gordon Wood and Wilfred Wilson — were attendants on him, suggesting that he might not have been an attendant but at the same level as Lady Cynthia and the Princess of Pless.&lt;ref name=&quot;:8&quot; /&gt; (34, Col. 3a; print p. 50)

==== The Structure of the Procession ====
In the visualization in the ''Morning Post'', Lady Cynthia's name is first, followed by the Princess of Pless, at the same level. Then the suites of ladies and gentlemen follow, indented to show they accompanied either the Princess of Pless or, perhaps, both Queens of Sheba. 

The ''Morning Post'' illustrated the procession like this:
# [[Social Victorians/People/Feversham|Lady Cynthia Graham]], as the Queen of Sheba
# Daisy (Mary Theresa) [[Social Victorians/People/Cornwallis-West|Cornwallis-West]], [[Social Victorians/People/Pless|Princess Henry of Pless]], as the Queen of Sheba
# The Suite of Ladies following Daisy, Princess of Pless (or perhaps both the two Queens of Sheba)
## Miss West: Miss Cornwallis West, [[Social Victorians/People/Cornwallis-West|Shelagh Cornwallis-West]], as her &quot;Ethiopian attendant&quot;&lt;ref name=&quot;:1&quot; /&gt;{{rp|p. 5, Col. 7c}}
## Miss Mary [[Social Victorians/People/Goelet|Goelet]]
## [[Social Victorians/People/Westminster#Lady C. Grosvenor|Lady C. Grosvenor]]
## Miss [[Social Victorians/People/Oppenheim|Rosalinda Oppenheim]]
# The Suite of Men following Daisy, Princess of Pless (less likely both the two Queens of Sheba)
## The [[Social Victorians/People/Keppel|Hon. George Keppel]], as King Solomon, in the Suite of Men following the two Queens of Sheba (Lady Cynthia Graham and Princess Henry of Pless)&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:0&quot; /&gt;{{rp|p. 7, Col. 5b}}
## [[Social Victorians/People/Sarah Spencer-Churchill Wilson|Wilfred Wilson]]
## [[Social Victorians/People/Portman|Arthur Portman]]
##[[Social Victorians/People/Halifax|Gordon Wood]]
## The [[Social Victorians/People/Bourke#Hon. Algernon Bourke|Hon. A. Bourke]]: Hon. Algernon Bourke (listed as being in this procession, but dressed as Isaac Walton, probably not; Gwendolen Bourke, however, as Salambo, likely was)
## [[Social Victorians/People/Cornwallis-West#George Cornwallis-West|George Cornwallis-West]], not mentioned in any newspaper report and by his own account in the court of his sister, Daisy, Princess of Pless.
# [[Social Victorians/People/Duncombe|Lady Alicia Duncombe]], as a Greek Slave
# [[Social Victorians/People/Bourke#Hon. Guendoline Bourke|Hon. Mrs. A. Bourke]]: Hon. Guendoline Bourke, as Salambo
# Minnie Paget, [[Social Victorians/People/Paget Family|Mrs. Arthur Paget]], as Cleopatra
# Gerald [[Social Victorians/People/Paget Family|Paget Paget]], likely Gerald Cecil Stewart Paget, as Marc Antony
#[[Social Victorians/People/Churchill#Jennie (Lady Randolph) Churchill|Lady Randolph Churchill]], according to the London Evening ''Mail''&lt;ref name=&quot;:9&quot; /&gt;{{rp|p. 8, Col. 1c}}, as Empress Theodora of Byzantium
# Lady [[Social Victorians/People/de Trafford|Violet de Trafford]]
#Alexandra Harriet Paget, [[Social Victorians/People/Colebrooke|Lady Colebrooke]]
# Two women walking together for some reason
## Hon. Mrs. Julia [[Social Victorians/People/Peel Family|Peel Maguire]] (the ''Morning Post'' has her both in the Oriental and the Duchesses processions)
## Miss [[Social Victorians/People/Muriel Wilson#Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball|Muriel Wilson]], as Queen Vashti
# [[Social Victorians/People/Fraser#Helena Violet Alice Fraser (Miss Keith Fraser)|Miss Keith Fraser]]
# Mary Charteris, [[Social Victorians/People/Charteris|Lady Elcho]]
# Mrs. [[Social Victorians/People/Hope-Vere|Hope-Vere]], as Medusa

The ''Morning Post'' typeset a visualization of the procession, more or less, like this:
   Lady Cynthia Graham .................. Queen of Sheba.
   Princess Pless ....................... Queen of Sheba.
           Miss West .................. )
           Miss Goelet ................ )
           Lady C. Grosvenor .......... ) Suite of Ladies.
           Miss Oppenheim ............. )
           The Hon. G. Keppel ......... )
           Wilfred Wilson  ............ )
           Arthur Portman ............. ) Suite of Men.
           Gordon Wood ................ )
           The Hon. A. Bourke ......... )
   Lady Alicia Duncombe ................  Greek Slave.
   Hon. Mrs. A. Bourke .................. Salambo.
   Mrs. Arthur Paget .................... Cleopatra.
   Gerald Paget Paget ................... Marc Antony.
   Lady De Trafford
   Hon. Mrs. Maguire ................   )
   Miss Muriel Wilson ................  )
   Miss Keith Fraser
   Lady Elcho
   Mrs. Hope-Vere
The ''Times'' article lists the members of this procession in paragraph form, but they are the same people in the same order in the list:&lt;blockquote&gt;Lady Cynthia Graham, Queen of Sheba; Princess Pless, Queen of Sheba; Miss West, Miss Goelet, Lady C. Grosvenorm, Miss Oppenheim, suite of ladies; Hon. G. Keppel, Wilfred Wilson, Arthur Postman, Gordon Wood, Hon. A Bourke, suite of men; Lady Alicia Dduncombe, Greek slave; Hon. Mrs. A. Bourke, Salambo; Mrs. Arthur Paget, Cleopatra; Gerald Paget Paget, Marc Antony; Lady Randolph Churchill, Lade de Trafford, Lady Colebrooke, Hon. Mrs. Maguire, Miss Muriel Wilson, Miss Keith Fraser, Lady Elcho, Mrs. Hope-Vere.&lt;ref name=&quot;:5&quot; /&gt; (12, Col. 1c)&lt;/blockquote&gt;

===Goddesses===
The women who walked in the procession of goddesses included these women listed in the ''Morning Post'' list:
*[[Social Victorians/People/Gerard|Lady Mary Gerard]] (at 256), as Astarte, Goddess of the Moon
*Sybil Vane, [[Social Victorians/People/Westmorland|Countess of Westmorland]] (at 219), as Hebe
*[[Social Victorians/People/Lurgan|Lady Emily Lurgan]] (at 56), a Fury
*[[Social Victorians/People/Cadogan|Lady Sophie Scott]] (at 57), a Fury
*[[Social Victorians/People/Shrewsbury|Mrs. Talbot]] (probably not the wife of [[Social Victorians/People/Talbot|Edmund Talbot]], as two other Mrs. Talbots were there, both wives of higher ranking men)
*Miss de Brienen (at 259)
*[[Social Victorians/People/Leslie|Mrs. Leonie Leslie]] (at 260) went as Brunhilde in the Goddesses procession
The ''Gentlewoman'' says, &quot;Among the gods and goddesses was Titania, the Queen of the Fairies; Lady Westmorland who made the prettiest Hebe; the Furies, Lady Lurgan and Lady Sophie Scott; and Lady Archibald Campbell, who elected to appear as Diana.&quot;&lt;ref name=&quot;:8&quot; /&gt;{{rp|p. 32, Col. 2c}}

Others dressed as goddesses   — likely dressed as individuals and not part of an organized group — include the following:
*Alice [[Social Victorians/People/Borthwick|Borthwick, Lady Glenesk]] (at 88) as Egeria (although Algernon Borthwick, Baron Glenesk did walk in the Elizabethan procession)
*Mrs. [[Social Victorians/People/Fanny Ronalds|Fanny Ronalds]] (at 92), as Euterpe, Goddess of Music
*[[Social Victorians/People/Argyll|Lady Archibald Campbell]] (at 377), as Artemis, goddess of the chase
*Mrs. Susannah [[Social Victorians/People/Arthur Stanley Wilson|Wilson Graham Menzies]] (at 378), as Titania.
*Lady A. Stanley: Lady Alice Maud Olivia [[Social Victorians/People/Stanley|Montagu Stanley]] (at 157) as Diana.

The ''Morning Post'' typeset a visualization of the procession, more or less, like this&lt;ref name=&quot;:0&quot; /&gt;{{rp|p. 7, Col. 5B}}:
 Goddesses were:
            Lady Gerard.
            Lady Westmorland.
            Lady Lurgan.
            Lady S. [Sophie?] Scott.
            Mrs. Talbot.
            Miss de Brienen.
            Mrs. Leslie.

===The Duchesses Procession===
The members of this procession included the following:
# Katherine Osborne, the [[Social Victorians/People/Leeds|Duchess of Leeds]] (at 35), as the fictional Persian character Lalla Rookh. While the ''Morning Post'' says she walked in the Duchesses Procession, she might have walked in the &quot;Oriental&quot; one instead.
# [[Social Victorians/People/Dudley|Lady Dudley]]: Georgina, Dowager Countess of Dudley (at 198)
# [[Social Victorians/People/Ripon|Gwladys, Countess de Grey]] (at 136), possibly as Cleopatra (according to the ''Carlisle Patriot'', she headed the &quot;Oriental&quot; procession, but the ''Morning Post'' visualization puts her with the Duchesses)&lt;ref&gt;&quot;Fancy Dress Ball: Unparalleled Splendour.&quot; ''Carlisle Patriot'' Friday 9 July 1897: 7 [of 8], Col. 4a–b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000365/18970709/084/0007.&lt;/ref&gt;
#[[Social Victorians/People/Churchill|Lady Randolph Churchill]] (at 132), as Empress Theodora of Byzantium
# Mrs. Maguire (the ''Morning Post'' has her both in the Oriental and the Duchesses processions)
# [[Social Victorians/People/Adele Grant Capell|Adele Grant Capell, Countess Essex]] (at 194)
# [[Social Victorians/People/Asquith|Margot Asquith]] (at 217), as a snake charmer
# Mrs. Leo (at 246)
# [[Social Victorians/People/Gosford|Louisa Acheson, Lady Gosford]] (at 140), as a lady in Charles V.'s Court(?)
# Edith Glyn, [[Social Victorians/People/Wolverton|Baroness Wolverton]] (at 130), as Britannia
# [[Social Victorians/People/Stanley|Lady Alice Maude Olivia Montagu Stanley]] (at 157) (but she seems more likely to have walked in the Goddess procession)
# Mr. [[Social Victorians/People/Brassey|L. Brassey]] (at 252), as Apollo
# [[Social Victorians/People/Gosford|Lady A. Acheson]] — Lady Alexandra Louise Elizabeth Acheson — (at 254), in Hunting Costume, period of Louis XV
# [[Social Victorians/People/Gosford|Lord Acheson]], Archibald Charles Montagu Brabazon Acheson (at 255), as Mignon Henri III. or Raoul di Nangis
# Lady J. Stanley (at 250)
# W. Stanley: [[Social Victorians/People/Stanley#Lord William Stanley and Lady Alexandra Stanley|Hon. Frederick William Stanley]] (at 473), in hunting dress (period of Louis XVI) or as Chasseur à Louis XV

The ''Morning Post'' typeset a visualization of the procession, more or less, like this, suggesting that everybody except Lady de Grey and Lady Wolverton was walking or perhaps dancing side by side in pairs.
               Duchess of Leeds.                 Lady Dudley.
                                  Lady de Grey.
               Lady Randolph Churchill.          Lady Colebrooke.
               Mrs. Maguire.                     Lady Essex.
               Mrs. Asquith.                     Mrs. Leo.
               Lady Gosford.                     E. Stanley.
                                 Lady Wolverton.
               Lady A. Stanley.                  L. Brassey.
               Lady A. Acheson.                  Lord Acheson.
               Lady J. Stanley.                  W. Stanley.

===Italian Procession===
The Venetians Procession is variously called the Venetian or 17th-century or Italian Procession or Quadrille in the newspapers. This group is also made up of subgroups: the Italian Procession, the Venetians and the 17th Century, each smaller procession with its own leader.

The ''Westminster Gazette'' says, &quot;The Venetian group might indeed have been called a 'dream of fair women,' as it numbered more decidedly beautiful women than any other at the ball.&quot;&lt;ref&gt;“The Duchess’s Costume Ball.” ''Westminster Gazette'' 03 July 1897 Saturday: 5 [of 8], Cols. 1a–3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002947/18970703/035/0005.&lt;/ref&gt;{{rp|p. 5, Col. 1}}

The ''Gentlewoman'' says,&lt;blockquote&gt;The Venetian Court was most picturesque, led by the Duchess of Portland, who looked magnificent in white brocade embroidered with silver, a diamond crown, and ropes of diamonds and pearl, round her neck. One of the most noticeable ladies of her Court was Lady Mar and Kellie, in white and green and silver, embroidered with gold.&lt;ref name=&quot;:8&quot; /&gt;{{rp|p. 32, Col. 3a}}&lt;/blockquote&gt;''Truth'' says something quite similar:&lt;blockquote&gt;The Venetian group was highly picturesque. Lord Lathom was Doge, and among the ladies and gentlemen of Venice were the Duchess of Portland, Countess of Mar and Kellie, Lady Alington, Mr. and Mrs. W. H. Grenfell, and Lord and Lady St. Oswald.&lt;ref name=&quot;:10&quot;&gt;“Girls’ Gossip.” ''Truth'' 8 July 1897, Thursday: 41 [of 70], Col. 1b – 42, Col. 2c. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0002961/18970708/089/0041.&lt;/ref&gt;{{rp|41, Col. 2c}}&lt;/blockquote&gt;

These are the names in the visualization in the ''Morning Post'' story.

==== Italian Procession ====

# The [[Social Victorians/People/Mar and Kellie|Countess Mar and Kellie]] (at 160), as Beatrice
# Lloyd Tyrell-Kenyon, 4th [[Social Victorians/People/Lloyd Kenyon|Baron Kenyon]] (at 167) as Guido Cavalcanti
# Mabel Winn, [[Social Victorians/People/Saint Oswald|Lady St. Oswald]] (at 284), as Duchessa di Caluria in the Italian procession or a Venetian lady of the 14th century
# Mr. [[Social Victorians/People/Wyndham|George Wyndham]] (at 221), as Signor di Samare
# [[Social Victorians/People/Forbes|Miss Blanche Forbes]] (at 285), as Donna Lucrezia Arcella
# Mr. [[Social Victorians/People/Schreiber|Schreiber]] (at 286), as Duca d'Iripolda
# [[Social Victorians/People/Higgins|Mrs. Higgins]] (at 287), as Donna Valeria Bodessa
# Mr. [[Social Victorians/People/Grenfell|William Henry Grenfell]] (at 222), as Signor di Argentina or as [[Social Victorians/People/Grenfell|Mercutio]]
# Mrs. Mary [[Social Victorians/People/Von Andre|Von André]] (at 289), as Desdemona
# Mr. [[Social Victorians/People/Walter Murray Guthrie|Murray Guthrie]] (at 290), as Otello
# [[Social Victorians/People/Montagu|Lady Alice Montagu]] (at 292), as Laura and escorted by Giles Fox-Strangways
# Giles Fox-Strangways (at 78), [[Social Victorians/People/Ilchester|Lord Stavordale]], as Petrarch
# Miss [[Social Victorians/People/Arthur Stanley Wilson#Enid Wilson|Enid Wilson]] (at 293), as Giulietta
# Lord Hyde: George Herbert Hyde [[Social Victorians/People/Villiers|Villiers]] (at 294), as Romeo

==== Venetians ====

# The [[Social Victorians/People/Lathom|Earl of Lathom]] (at 125), as Il Doge, Giovannino de Medici
#Mr. [[Social Victorians/People/Williams#Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball|Hwfa Williams]] (was unable to attend), as Cardinale Giovanni Bembo
#Mrs. [[Social Victorians/People/Williams#Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball|Hwfa Williams]] (was unable to attend), as Caterina Cornaro (Regina di Cipri)
# [[Social Victorians/People/Guest|Hon. Ivor Guest]] (at 295), as Marco (Re di Cipri)
# Mildred Cadogan, [[Social Victorians/People/Cadogan|Viscountess Chelsea]] (at 162), Venditrice di Fiori, a Veronese lady
# [[Social Victorians/People/Sarah Spencer-Churchill Wilson|Mr. Clarence Wilson]] (at 300), as Buffone
#[[Social Victorians/People/Fortescue|Hon. Seymour Fortescue]] (at 296), as Avocato
#[[Social Victorians/People/Warwick#Hon. Sir Sidney Robert Greville|Hon. S. Greville]] (at 297), as Cipriano
# The [[Social Victorians/People/Curzon|Hon. Mrs. George (Mary) Curzon]] (at 301), as Marchesa Malaspina
# [[Social Victorians/People/Peel Family#Hon. George Peel|Hon. George Peel]] (at 302), as Luigi Giorgi
# Ettie (Mrs. W.) [[Social Victorians/People/Grenfell|Grenfell]] (at 200), as Contessa Maria Cicogna or Maria de Medici
# The [[Social Victorians/People/Charteris|Hon. Evan Charteris]] (at 303), as Cavaliere Vittorio
# [[Social Victorians/People/Grosvenor|Lady Lettice Grosvenor]] (at 304), as Bianca Capelli
# Lord [[Social Victorians/People/Thynne|Alexander Thynne]] (at 305), as Marino Grimani
# Mrs. [[Social Victorians/People/Portland|Cavendish Bentinck]] (at 264), as Grandezza degli Antenati
# [[Social Victorians/People/Lurgan#Hon. Cecil Brownlow|Hon. Cecil Brownlow]] (at 305), as Nicolo Danabi
# Mrs. [[Social Victorians/People/Walter Murray Guthrie|Olive Guthrie]] (at 291), as Marguerita Grimani
# [[Social Victorians/People/Sarah Spencer-Churchill Wilson|Mr. Herbert Wilson]] (at 307), as Antonio Priali (misspelled as Briali)

==== 17th-Century Procession ====

# [[Social Victorians/People/Arthur Sassoon|Louise (Mrs. Arthur) Sassoon]] (at 202), as La Dogaressa, led the 17th-century procession, with two nephews as attendants:
##[[Social Victorians/People/Rothschild Family|Evelyn Achille de Rothschild]] (at 669), as a page to the Doge's Wife
##[[Social Victorians/People/Rothschild Family|Anthony Gustav de Rothschild]] (at 670), as a page to the Doge's Wife
# Arthur Wellesley Peel, 1st [[Social Victorians/People/Peel Family|Viscount Peel]] (at 74), as Il Doge
# Winifred, [[Social Victorians/People/Portland|Duchess of Portland]] (at 29), as Duchessa di Savoia
# William, [[Social Victorians/People/Portland|Duke of Portland]] (at 28), as Duca Filiberto di Savoia (or possibly the Duke of Buckingham)
# [[Social Victorians/People/Feversham|Lady Helen Vincent]] (at 215), as Contessa Valentina Gateago
# [[Social Victorians/People/Feversham|Sir Edgar Vincent]] (at 226), as II Conte Oravio or Orayio
# Mrs. Gerard Leigh (at 308), as Lucrezia de Rossi
# [[Social Victorians/People/Higgins|Mr. Higgins]] (at 288), as Sanchio di Sedilla
# Mrs. [[Social Victorians/People/Drummond|Katherine Mary Drummond]] (at 309), as Donna Caranado
# [[Social Victorians/People/Henry White|Mr. Henry White]] (at 310), as Giovanni Felici (or possibly Henri de Lorraine, Duc de Guise)
# Miss [[Social Victorians/People/Mildred Grenfell|Mildred Grenfell]] (at 30), as Bianca di Piacoma, accompanying Winifred, [[Social Victorians/People/Portland|Duchess of Portland]]
# Mr. Norton (at ), as Guyman di Silva (the ''Times'' and hence the Evening ''Mail'' says Morton)
# [[Social Victorians/People/Fraser#Captain Hugh Fraser|Captain Fraser]] (at 244), as Duca di Tarsis
Visitors to the Court of Savoia
# Windham, [[Social Victorians/People/Dunraven|Earl of Dunraven]] (at 199), as Cardinal Mazzarin
# Consuelo, [[Social Victorians/People/Manchester|Duchess of Manchester]] (at 175), as Anne d'Autriche [this isn't right: she's in the Russian Procession with the Duke of Marlborough, as the French Ambassador to the Court of Catherine II and his wife.]
# Mr. [[Social Victorians/People/Beraud|Jean Béraud]] (at 312), as Cinq Mars

==== Not Listed in the ''Morning Post'' story, But Still ====
... said somewhere to have been in the Italian Procession or might logically have processed with them.
#[[Social Victorians/People/Salisbury|Lady Edward Cecil]], probably Violet Georgina Maxse Gascoyne-Cecil (at 102)

==== The Morning Post Visualization ====
In the ''Morning Post'' visualization of the procession, Beatrice, the Countess of Mar and Kellie led the procession. The typeset visualization looks, more or less, like this:
                                  ITALIAN PROCESSION
                  ''Beatrice'',                            ''Guido Cavalcanti'',
     The Countess of Mar and Kellie.                      Lord Kenyon.
             ''Duchessa di Caluria'',                      ''Signor di Samare'',
               Lady St. Oswald.                       Mr. George Wyndham.
           ''Donna Lucrezia Arcella'',                      ''Duca d'Iripolda'',
            Miss Blanche Forbes.                         Mr. Schreiber.
           ''Donna Valeria'' ''Bodessa'',                     ''Signor di Argentina'',
               Mrs. Higgins.                            Mr. W. Grenfell.
                 ''Desdemona'',                                 ''Otello'',
              Mrs. Von André.                         Mr. Murray Guthrie.
                  ''Laura'',                                  ''Petrarch'',
           Lady Alice Montagu.                         Lord Stavordale.
                ''Giulietta'',                                 ''Romeo'',
            Miss Enid Wilson.                             Lord Hyde.
 
                                     Venetians
        ''Il Doge (Giovannino de Medici)'',                ''Marco (Redi Cipri)'',
             The Earl of Lathom.                        Hon. Ivor Guest.
         Avocato,                Venditrice di Fiori,             Cipriano,
  Hon. Seymour Fortescue         Viscountess Chelsea.        Hon. S. Greville.
                                     Buffone,
                               Mr. Clarence Wilson.
             ''Marchesa Malaspina'',                         ''Luigi Giorgi'',
          Hon. Mrs. George Curzon.                     Hon. George Peel.
            ''Contessa Maria Cicogna'',                        ''Cavaliere Vittorio'',
              Mrs. W. Grenfell.                           Hon. Evan Charteris.
               ''Bianca Capelli'',                              ''Marino Grimani'',
          Lady Lettice Grosvenor.                       Lord Alexander Thynne. 
         ''Grandezza degli Autenati'',                          ''Nicolo Danabi'',
          Mrs. Cavendish Bentinck.                       Hon. Cecil Brownlow.
            ''Marguerita Grimani''.                             ''Antonio Priali'',
               Mrs. Guthrie.                              Mr. Herbert Wilson.
 
                                   17th CENTURY
               ''La Dogaressa'',                                  ''Il Doge'',
            Mrs. Arthur Sassoon.                           Viscount Peel.
             ''Duchessa di Savoia'',                      ''Duca Filiberto di Savoia'',
          The Duchess of Partland.                       The Duke of Portland.
         ''Contessa Valentina Gateago'',                        ''II Conte Oravio'',
             Lady Helen Vincent.                           Sir Edgar Vincent.
              ''Lucrezia de Rossi'',                          ''Sanchio di Sedilla'',
              Mrs. Gerard Leigh.                              Mr. Higgins.
                ''Donna Caranado'',                             ''Giovanni Felici'',
                Mrs. Drummond.                                Mr. H. White.
              ''Bianci di Piacoma'',                            ''Guyman di Silva'',
            Miss Mildred Grenfell.                             Mr. Norton.
                                   ''Duca di Tarsis'',
                                   Captain Fraser.
                           Visitors to the Court of Savoia.
       ''Cardinal Mazzarin'',         ''Anne d'Autriche'',                ''Cinq Mars'',
     The Earl of Dunraven.    The Duchess of Manchester.       Mr. Jean Bérand.

===Austrian===
The ''Gentlewoman'' says, &quot;The Austrian Court was a wonderful procession, headed by the Marchioness of Londonderry as the Empress Marie Thérèse. She wore the famous Londonderry diamonds, which included a diamond crown copied exactly from one worn by the Empress Marie Thérèse on her powdered hair. She was followed by four young Archduchesses, in white and silver and pale blue ribbons.&quot;&lt;ref name=&quot;:8&quot; /&gt;{{rp|p. 32, Col. 2c–3a}} 

The Austrian procession and quadrille were headed up by Theresa Vane-Tempest-Stewart, [[Social Victorians/People/Londonderry|Marchioness of Londonderry]] as Marie Thèrése of Austria. Most accounts say it had 4 archduchesses in attendance, even though the Guernsey ''Star'' reported 5 and the ''Belfast News-Letter'' adds [[Social Victorians/People/Somerset|Miss Seymour]] (at 406): 

* &quot;four beautiful young Archduchesses in white and silver with pale blue ribbons, and wearing white plumes in their powdered hair&quot;&lt;ref name=&quot;:1&quot;&gt;&quot;Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses.&quot; ''London Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and http://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.&lt;/ref&gt;{{rp|p. 5, Col. 7A}}
*&quot;four young Archduchesses, in white and silver and pale blue ribbons. These ladies were impersonated by Lady Helen Stewart, Lady Beatrice Butler, Lady Beatrix FitzMaurice, and Lady Alexandra Hamilton.&quot;&lt;ref name=&quot;:8&quot; /&gt;{{rp|p. 32, Col. 2c–3a}}
* &quot;five Archduchesses and five Archdukes. The former, all attired exactly alike in white and silver brocade, were Lady Helen Stewart, Lady Beatrice Butler, Lady Beatrix FitzMaurice, Lady Alexandra Hamilton, and Miss Stirling.&quot;&lt;ref name=&quot;:3&quot;&gt;&quot;Duchess of Devonshire's Fancy-Dress Ball. Brilliant Spectacle.&quot; The [Guernsey] ''Star'' 6 July 1897, Tuesday: 1 [of 4], Col. 1–2. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000184/18970706/003/0001.&lt;/ref&gt;

Possibly this quite-large group had subsections: the ''Morning Post'' and the ''Times'' mention for example &quot;the Archduchess Marie-Karoline and Emperor Joseph II section of the Austrian Court of Maria Theresa Quadrille.&quot;&lt;ref name=&quot;:0&quot; /&gt;{{rp|p. 7, Col. 6b}} &lt;ref name=&quot;:5&quot;&gt;&quot;Ball at Devonshire House.&quot; ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt;
# Section 1
## Theresa Vane-Tempest-Stewart, [[Social Victorians/People/Londonderry|Marchioness of Londonderry]] (at 42), as Marie Thérèse of Austria
## Henry Petty-FitzMaurice, [[Social Victorians/People/Lansdowne|Marquis of Lansdowne]] (at 52), as Prince Kaunitz
## Maud Hamilton Petty-Fitzmaurice, [[Social Victorians/People/Lansdowne|Marchioness of Lansdowne]] (at 51), as Lady Keith, wife of the British Ambassador at the Court of Marie Thérèse
##[[Social Victorians/People/Winchester|Augustus, Marquis of Winchester]], as a Coldstream Guard at Vienna
# Section 2: Archduchess Marie-Karoline and Emperor Joseph II section of the Austrian Court of Maria Theresa Quadrille
## Archduchesses
###[[Social Victorians/People/Ormonde|Lady Beatrice Butler]] (at 45), Archduchess Marie-Karoline
###[[Social Victorians/People/Abercorn|Lady Alexandra Hamilton]] (at 46), Archduchess Marie-Josepha
###[[Social Victorians/People/Lansdowne|Lady Beatrix Petty-FitzMaurice]] (at 44), as Archduchess Marie Anna
### Lady Helen Stewart ([[Social Victorians/People/Londonderry|Vane-Tempest-Stewart]]) (at 43), as the Archduchess Marie Christine of Austria
## Charles Stewart Henry Vane-Tempest-Stewart, [[Social Victorians/People/Londonderry|Viscount Castlereagh]] (at 73), attended as Maria Thérèse's son Emperor Joseph II (Charles Vane-Tempest-Stewart, Viscount Castlereagh, was the [[Social Victorians/People/Londonderry|Marchioness of Londonderry]]'s son.)
## Mr. [[Social Victorians/People/Gathorne-Hardy|Gathorne-Hardy]] (at 352), as Archduke Leopold
## Charles William Reginald Duncombe, [[Social Victorians/People/Helmsley|Viscount Helmsley]] (at 353), as Archduke Charles
##[[Social Victorians/People/Lurgan|Lord William Lurgan]] (at 165), as Duke Albert von Sachsen-Texhen or Sachsentexhen
# Small Group of 4
## [[Social Victorians/People/Magheramorne|Lady Magheramorne]] (at 355), as Maria-Amelia, Princess of Lorraine
## Lady [[Social Victorians/People/Beaumont|Aline Beaumont]] (at 356), as as Marie Josephe of Austria or Queen of Sardinia
## Lord Ava: Archibald Hamilton-Temple-Blackwood, [[Social Victorians/People/Hamilton Temple Blackwood#Archibald Hamilton-Temple-Blackwood, Earl of Ava|Earl of Ava]] (at 357), as Archduke Maximilian
## Mr. [[Social Victorians/People/Ancaster|C. Willoughby]] (at 358), as Grand Duke Charles of Tuscany
# Small Group of 4
## Mrs. [[Social Victorians/People/Grimthorpe#Mr. and Mrs. Gervase BeckettMr. and Mrs. Gervase Beckett|Mabel (G. Gervase) Beckett]] (at 359), as Princess Elenora of Lichtenstein
## Siegfried, [[Social Victorians/People/Clary Aldringen|Count Clary]] (at 205), as General Count Nadasdy
## Mrs. [[Social Victorians/People/Grimthorpe#Mr. and Mrs. R. Beckett|Muriel Beckett]] [sic Mrs. R. Beckeet] (at 482), as Princess Isabella of Parma
## [[Social Victorians/People/Hadik|Count Hadik]] (at 361), as Field-Marshal Hadik
# Small Group of 4
## Cicely Gascoyne-Cecil, [[Social Victorians/People/Salisbury|Viscountess Cranborne]] (at 196), as Princess Josepha of Bavaria
## Schomberg McDonnell, [[Social Victorians/People/Antrim|Mr. Schomberg M'Donnell]] (at 104), as Duke Ferdinand of Modena
## [[Social Victorians/People/Midleton|Lady Hilda Charteris Brodrick]] (at 362, Lady H. Brodrick), as Princess Marie Künigunde of Saxony
## Mr. Jack [[Social Victorians/People/Arthur Stanley Wilson|Graham Menzies]] (at 362), as Freiherr von Bartenstein
# Small Group of 4
## Mrs. [[Social Victorians/People/William James|Evelyn James]] (at 364), as Archduchess Elizabeth
## [[Social Victorians/People/Lansdowne|Lady C. FitzMaurice]] (at 365), as Secretary to Kaunitz, personated as listed above by Henry Petty-FitzMaurice, [[Social Victorians/People/Lansdowne|Marquis of Lansdowne]] (at 558)
## Muriel Duncombe, [[Social Victorians/People/Helmsley|Viscountess Helmsley]] (at 354), as Princess Charlotte of Lorraine
## Henry Petty-FitzMaurice, [[Social Victorians/People/Kerry|Earl of Kerry]] (at 72), as Count Mercy d'Argentau
# Small Group of 4
## [[Social Victorians/People/Cavendish|Lady E. Cavendish]] (probably Lady Evelyn Cavendish, at 164), as Countess Trautmannsdorf
##[[Social Victorians/People/Mildmay|Mr. F. B. Mildmay]] (at 95), as &quot;Field-Marshal Count Charles of Batthyany&quot;
## (Lady M.) [[Social Victorians/People/Cavendish|Lady Moyra Cavendish]] (at 366)
## James Somerville, [[Social Victorians/People/Athlumney|Lord Athlumney]] (at 367), as Prince Metternich
# Small Group of 2? (or they belong above, her with the Archduchesses?)
##[[Social Victorians/People/Stirling|Miss Stirling]] (at 47), as Countess Kinskey in the Austrian Court of Maria Theresa Quadrille
## [[Social Victorians/People/Midleton#St. John Broderick|Mr. St. John Brodrick]] (at 368), as Count Kinskey
# People not in the ''Morning Post'' visualization but elsewhere said to have been in this procession:
## Charles Vane-Tempest-Stewart, the [[Social Victorians/People/Londonderry|Marquess of Londonderry]] (at 511), in the procession according to the ''Morning Post'' story
## [[Social Victorians/People/Lansdowne|Edmond Fitzmaurice]] (at 627), a Courtier of the Empress Marie Thérèse
##Miss [[Social Victorians/People/Ellis|Alexandra Ellis]] (at 655), in a Thérèse costume

The ''Morning Post'' typeset a visualization of the procession, more or less, like this&lt;ref name=&quot;:0&quot; /&gt;{{rp|p. 7, Col. 6b}}:
 Lady Londonderry . . . . . Empress Maria Theresa.
 Lord Lansdowne . . . . . . Prince Kaunitz.
 Lady Lansdowne . . . . . . Lady Keith.
 Lord Winchester  . . . . . A Coldstream Guard at Vienna.
 
 Lady B. Butler . . . . . . Archduchess Marie-Karoline.
 Lord Castlereagh . . . . . Emperor Joseph II.
 Lady A. Hamilton . . . . . Archduchess Marie-Josepha.
 Mr. Gathorne-Hardy . . . . Archduke Leopold.
 Lady B. FitzMaurice  . . . Archduchess Marie Anna.
 Lord Helmsley  . . . . . . Archduke Charles.
 Lady Helen Stewart . . . . Archduchess Marie Christine. 
 Lord Lurgan  . . . . . . . Duke Albert von Sachsen-Texhen. 
 
 Lady Magheramorne . . . . . Maria-Amelia, Princess of Lorraine. 
 Lady Aline Beaumont . . . . Queen of Sardinia. 
 Lord Ava  . . . . . . . . . Archduke Maximilian. 
 Mr. C. Willoughby . . . . . Grand Duke Charles of Tuscany. 
 
 Mrs. G. Beckett . . . . . . Princess Elenora of Lichtenstein. 
 Count Clary . . . . . . . . General Count Nadasdy. 
 Mrs. R. Beckeet [sic] . . . Princess Isabella of Parma. 
 Count Hadik . . . . . . . . Field-Marshal Hadik. 
 
 Lady Cranborne  . . . . . . Princess Josepha of Bavaria. 
 Mr. M'Donnel  . . . . . . . Duke Ferdinand of Modena. 
 Lady H. Brodrick  . . . . . Princess Marie Künigunde of Saxony. 
 Mr. Menzies . . . . . . . . Freiherr von Bartenstein.
 
 Mrs. James  . . . . . . . . Archduchess Elizabeth. 
 Lady C. FitzMaurice . . . . Secretary to Kaunitz. 
 Lady Helmsley . . . . . . . Princess Charlotte of Lorraine. 
 Lord Kerry  . . . . . . . . Count Mercy d'Argentau. 
 
 Lady E. Cavendish . . . . . Countess Trautmannsdorf. 
 Mr. Mildmay . . . . . . . . Field-Marshal Count Charles of Batthyany. 
 Lady M. Cavendish . . . . . Countess Lützau (A Lady-in-Waiting to Maria Theresa). 
 Lord Athlumney  . . . . . . Prince Metternich. 
 
 Miss Stirling . . . . . . . Countess Kinskey. 
 Mr. Brodrick  . . . . . . . Count Philip Kinsky.

===Russian===
According to the ''Gentlewoman'',&lt;blockquote&gt;The Russian Court formed a dazzling procession, headed by Lady Raincliffe, as the Empress Catherine; her gown of white satin was studded with rubies, emeralds, and turquoises, and across her bodice she wore a blue ribbon with the Orders of the Star and Eagle, and upon her head a Russian crown of diamonds. Beside her was Prince Orloff, represented by Prince Henry Pless, in a costume of red cloth with heavy gold embroideries; he also wore the Order of St. Catherine. There were eight officers of the Imperial Court accompanying the Empress, whilst Lady Henry Bentinck and Lady Yarborough impersonated ladies of her suite, amongst which one of the most striking figures was Mr. Cresswell, as her Chamberlain, in a costume of cerise velvet, covered with the double-headed eagle of Russia in gold, which embroidery was repeated on his pink satin vest, his white satin breeches, and his silk stockings.&lt;ref name=&quot;:8&quot; /&gt;{{rp|p. 32, 3b}}&lt;/blockquote&gt;

''Truth'' says,&lt;blockquote&gt;Lady Raincliffe as Catherine of Russia was a marvel of millinery in yellow / and gold, ermine and rubies. Her lords and ladies emulated her splendour, and among the most successful were the Duchesses of Marlborough and Newcastle, Lady Yarborough, Lady Henry Bentinck, Lord Raincliffe, and Mr. Cresswall.&lt;ref name=&quot;:10&quot; /&gt;{{rp|41, Col 2c – 42, Col. 1a}}&lt;/blockquote&gt;
# The &quot;Trumpeters of the Imperial Guard&quot; led the procession
# The first group, if they did break into subgroups
## Lord Henry Bentinck (probably [[Social Victorians/People/Cavendish Bentinck|Lord Henry Cavendish Bentinck]]) (at 262), as Count Poneatowski (afterwards King of Poland)
## Grace Denison, [[Social Victorians/People/Londesborough|Viscountess Raincliffe]] (at 75), as Catherine II of Russia (after the picture by Lambi)
## [[Social Victorians/People/Heeren|Count Heeren]] (at 265), as Duc de Ligne
## [[Social Victorians/People/Cresswell|Addison Francis Baker-Cresswell]] (Mr. A. F. B. Cresswell) (at 103), as Count Lausköi, Chamberlain of the Empress Catherine II of Russia
# The second group
## [[Social Victorians/People/Pless|Prince Henry of Pless]] (at 40), as Count Orloff
## Mrs. H. T. [[Social Victorians/People/Barclay|Barclay]] (at 266), Princess Shakofsky
## Mr. [[Social Victorians/People/Myddleton-Biddulph|Biddulph]] (at 268), as Count Soltykoff
# The third group, with the 8 &quot;Imperial Guard&quot; walking along the outside (or at least typeset that way) of the &quot;Ladies and Gentlemen of the Court&quot;
## Imperial Guards: left side
### William Denison, [[Social Victorians/People/Londesborough|Viscount Raincliffe]] (at 76)
### Captain [[Social Victorians/People/Cook|E. B. Cook]] (at 269)
### Hon. [[Social Victorians/People/Dudley#Hon.%20Gerald%20Ernest%20Francis%20Ward|Gerald Ward]] (at  271)
###[[Social Victorians/People/Forbes|James Stewart Forbes]] (at 273)&lt;ref name=&quot;:1&quot; /&gt;{{rp|p. 5, Col. 7a}}
## Imperial Guards: right side
### Lord [[Social Victorians/People/Romilly|Romilly]] (at 269)
### Mr. H. T. [[Social Victorians/People/Barclay|Barclay]] (at 267)
### The Hon. [[Social Victorians/People/Campbell|Cecil Campbell]] (at 272)
### Mr. [[Social Victorians/People/Arthur Stanley Wilson|C. Wellesley Wilson]] (at 274)
## Ladies and Gentlemen of the Court
###[[Social Victorians/People/Consuelo Vanderbilt Spencer-Churchill|Consuelo Vanderbilt Spencer-Churchill]], the [[Social Victorians/People/Marlborough|Duchess of Marlborough]] (at 174)
### Sunny (Charles Richard John) Spencer-Churchill, the [[Social Victorians/People/Marlborough|Duke of Marlborough]] (at 142), one of the Gentlemen of the Court of Catherine II of Russia
### Kathleen Florence May Candy Pelham-Clinton, [[Social Victorians/People/Newcastle|Duchess of Newcastle]] (at 150)
### Charles Anderson-Pelham, [[Social Victorians/People/Yarborough|Earl of Yarborough]] (at 61), with Lady Yarborough, was among Ladies and Gentlemen of the Court
### Marcia Anderson-Pelham, [[Social Victorians/People/Yarborough|Countess of Yarborough]] (at 54)
### [[Social Victorians/People/Buchan|Lord Shipley Cardross]] (at 275)
### [[Social Victorians/People/Buchan|Lady Rosalie Cardross]] (at 276)
### [[Social Victorians/People/Stourton|Herbert Marmaduke Joseph Stourton]] (at 277)
### the [[Social Victorians/People/Buchan|Hon. M. (Muriel) Erskine]] (at 278), as La Marquise de Vintimille du Luc
### Mr. Elliot: [[Social Victorians/People/Minto|Sir Henry George Elliot]] (at 279)
### Lady Henry Bentinck ([[Social Victorians/People/Cavendish Bentinck|Lady Henry Cavendish Bentinck]]) (at 263)
### N. [[Social Victorians/People/Boulatzell|Boulatzell]] (at 280), as Prince of Mingrelia, one of the Gentlemen of the Court in the procession of the Empress Catherine II of Russia
### [[Social Victorians/People/Spicer|Lady Margaret Spicer]] (at 281), as Countess Soltykoff
### M. [[Social Victorians/People/Gourko|Nicholas Gourko]] (at 108)
###[[Social Victorians/People/Londesborough|Lady Mildred Denison]] (at 283)
### Charles Henry John Chetwynd-Talbot, 20th [[Social Victorians/People/Shrewsbury|Earl of Shrewsbury]] (at 101), as a member of the Court of Catherine II of Russia
### &quot;Black Attendants&quot;
The ''Guernsey Star'' suggests that the Hon. Mrs. Erskine's daughter was in this procession, but the ''Morning Post'' does not list her.&lt;ref name=&quot;:3&quot; /&gt; The [[Social Victorians/People/Durham#Hon. Cecil Lambton|Hon. Cecil Lambton]] (at 628) was also there, apparently, and sold his costume to theatre director Arthur Collins, who directed ''The White Heather''.&lt;ref&gt;&quot;The Morning’s News.&quot; London ''Daily News'' 18 September 1897, Saturday: 5 [of 8], Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970918/027/0005.&lt;/ref&gt;

The ''Morning Post''&lt;nowiki/&gt;'s visualization of the procession looks more or less like this&lt;ref name=&quot;:0&quot; /&gt;{{rp|p. 7, Col. 5b}}:
                              Trumpeters of the Imperial Guard.
 ''Count Poneatowski''            ''Empress Catherine II. of Russia''           ''Duc de Ligne,''
 ''(afterwards King'' ''of Poland),''  ''(after the picture by Lambi)'',            Count Heeren.           
 Lord Henry Bentinck.                 Lady Raincliffe.                  ''Count Lausköi'',
                                                                        Mr. Cresswell.
 ''Count Orloff'',                     ''Princess Shakofsky'',                ''Count Soltykoff'',
 Prince Henry of Pless.            Mrs. H. T. Barclay.                   Mr. Biddulph.
[&quot;Imperial Guard.&quot; — typeset vertically up the left and down the right side of the column, with 2 vertical rules separating the two columns of names.]
                  Lord Raincliffe.   |           |   Lord Romilly.
                     Captain Cook.   |           |   Mr. H. T. Barclay.
                 Hon. Gerald Ward.   |           |   Hon. Cecil Campbell.
                    Mr. J. Forbes.   |           |   Mr. T. W. Wilson.
                             Ladies and Gentlemen of the Court.
                  Duchess of Marlborough.      Duke of Marlborough.
                  Duchess of Newcastle.        Earl of Yarborough.
                  Countess of Yarborough.      Lord Cardross.
                  Lady Cardross.               Mr. Stourton.
                  Hon. M. Erskine.             Mr. Elliot.
                  Lady Henry Bentinck.         M. Botalzell.
                  Lady Margaret Spicer.        M. Gourko.
                  Lady Mildred Denison.        Earl of Shrewsbury.
                                    Black Attendants.

===Louis XV and XVI Period===
Louis XV was King of France 1715–1774, although his reign began when he reached maturity in 1724. Louis XVI reigned 1774–1792.

[[Social Victorians/People/Warwick|Daisy, Countess Warwick]], as Marie Antoinette, led this group rather than any of the Louis.&lt;ref&gt;&quot;Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses.&quot; ''London Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and http://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.&lt;/ref&gt;{{rp|p. 5, Col. 9c}} The Duchess of Devonshire may have asked her to do so, and this &quot;court&quot; may represent some part of the Countess of Warwick's social network.

Sounding as if the writer has confused Marie Antoinette with Queen Elizabeth I of the UK, ''Truth'' says,&lt;blockquote&gt;The Countess of Warwick, as Marie Antoinette, in white and blue, with golden fleur-de-lys upon her velvet train, was the centre of a picturesque group, among whom was the Earl of Essex, dressed as his ancestor of that period, and the Earl of Mar and Kellie as Sir Walter Raleigh.&lt;ref name=&quot;:10&quot; /&gt;{{rp|42, Col 1a}}&lt;/blockquote&gt;

The ''Morning Post'' calls this one a ''quadrille'' rather than a ''procession'', the quadrille of the Louis XV and XVI Period,&lt;ref name=&quot;:0&quot; /&gt;{{rp|7, Col. 6B}} but the distinction is probably not important. Generally, the courts processed in and then danced a quadrille before the Royals.
# Headed by [[Social Victorians/People/Warwick|Daisy, Countess Warwick]] (at 53) as Marie Antoinette&lt;ref name=&quot;:3&quot; /&gt;, as La Reine Marie Antoinette&lt;ref name=&quot;:0&quot; /&gt;
## Plus 4 boys dressed as pages&lt;ref&gt;“The Duchess of Devonshire’s Ball.” ''Chelmsford Chronicle''9 July 1897, Friday: 2 of 8. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000322/18970709/008/0002.&lt;/ref&gt;
# Georgiana Elizabeth Spencer-Churchill Curzon, [[Social Victorians/People/Howe|Viscountess Curzon]], as La Reine Marie Leszuiska in the quadrille of the period of Louis XV and XVI
# Nellie, [[Social Victorians/People/Kilmorey|Countess Kilmorey]], as Madame du Barry
# [[Social Victorians/People/Gordon-Lennox|Lady Algernon Gordon-Lennox]], as Princesse de Lamballe
# Lady [[Social Victorians/People/Burton#Lady Burton|Harriet Burton]], as Madame de Tençin
# Florence Canning, [[Social Victorians/People/Garvagh|Lady Garvagh]], as Comtesse d'Artois
# The Hon. Mrs. Greville, as Madame Elizabeth de France
#[[Social Victorians/People/Keppel|Alice (the Hon. Mrs. George) Keppel]], as Madame de Polignac
# Lady Rose Leigh, as Duchesse de Villars
# [[Social Victorians/People/Farquharson|Mrs. Farquharson]], as L'Archiduchesse Louise
# Miss [[Social Victorians/People/Naylor|Mary Naylor]], as Comtesse de Charny
# The [[Social Victorians/People/Sackville West|Hon. Mrs. Sackville West]], as Duchess of Dorset
# Henry Arthur Cadogan, [[Social Victorians/People/Cadogan|Viscount Chelsea]], as Le Roi Louis XV in the quadrille of the Louis XV. and Louis XVI. Period
# Lord Camden: John Pratt, [[Social Victorians/People/Camden|4th Marquess Camden]], as Duc de Richelieu
# The [[Social Victorians/People/Alington|Hon. Humphrey Sturt]], M.P., as an Abbé de l'Epoque
# Lord [[Social Victorians/People/Burton#Lord Burton|Michael Burton]], as Cardinal Dubois
# Mousquetaires et Militaires de l'Epoque
## Luke White, [[Social Victorians/People/Annaly|Lord Annaly]]
## Lord Tullibardine: John George Stewart-Murray, [[Social Victorians/People/Atholl|Marquess of Tullibardine]]
## Lord [[Social Victorians/People/Atholl|George Stewart-Murray]]
## Frederick Edward Guest, the [[Social Victorians/People/Guest|Hon. F. Guest]], as one of the Mousquetaires et Militaires de l'Epoque in the Quadrille of the Louis XV. and Louis XVI.
## [[Social Victorians/People/Cadogan|Sir Samuel Scott]], Bart., one of the Mousquetaires et Militaires de l'Epoque in the Quadrille of the Louis XV. and Louis XVI
## [[Social Victorians/People/Sarah Spencer-Churchill Wilson|Captain Gordon Wilson]], one of the Mousquetaires et Militaires de l'Epoque in the Quadrille of the Louis XV. and Louis XVI
## [[Social Victorians/People/Durham#Captain the Hon. W. Lambton|Captain the Hon. W. Lambton]], likely [[Social Victorians/People/Durham|Hon. Sir William Lambton]], one of the Mousquetaires et Militaires de l'Epoque in the Quadrille of the Louis XV. and Louis XVI.
## [[Social Victorians/People/Elliot|Captain Gilbert Elliot]]
## [[Social Victorians/People/Warwick|Mr. Frank Dugdale]]
##Carlo Ermes Visconti, [[Social Victorians/People/San Vito|Marchese di San Vito]], as one of the Mousquetaires
# [[Social Victorians/People/Muriel Wilson|Mr. Clive Wilson]], as Le Comte de Ferson in the Quadrille of the Louis XV. and Louis XVI
# Mr. [[Social Victorians/People/Lowe|W. M. Lowe]], as Gentilhomme de la Cour Louis XVI
# [[Social Victorians/People/Morley|Rt. Hon. Arnold Morley]], was dressed as a Gentilhomme de la Cour Louis XV in the Quadrille of the Louis XV. and Louis XVI. or Duc de Choiseul.
Not listed in the visualization in the ''Morning Post'' and probably not a member of the quadrille but mentioned elsewhere as having been dressed in a costume of this period:
# Millicent, [[Social Victorians/People/Sutherland|Duchess of Sutherland]], &quot;belonged to the Louis Seize group of the Countess of Warwick&quot;&lt;ref name=&quot;:3&quot; /&gt;
# The R[[Social Victorians/People/Chamberlain|ight Hon. Joseph Chamberlain]], as a gentleman of the Louis XVI period
#[[Social Victorians/People/Sarah Spencer-Churchill Wilson|Lady Sarah Wilson]], as Madame de Pompadour
# Mr. [[Social Victorians/People/Loder|Gerald Loder]], M.P., as a Gentleman of the Court of Louis XVI
#[[Social Victorians/People/Stanley|Sir George Frederick Stanley]], as Maro (period of Louis XVI)
#[[Social Victorians/People/Stanley|Lady Isobel Stanley]], in hunting costume, period of Louis XVI
#[[Social Victorians/People/Alva|Jacobo Fitz-James Stuart y Falcó, Lord Alva]], as as a courtier of Louis XV
#[[Social Victorians/People/Carrington|Charles Carington, the Earl Carington]], as Louis Seize
#Charles Chetwynd-Talbot, [[Social Victorians/People/Shrewsbury|Earl of Shrewsbury]], either a Gentleman of the Court of Lois XV. or a Gentleman of the Court of the Empress Catherine II of Russia

The ''Morning Post'' typeset a visualization of the procession, more or less, like this:
 Louis XV. and Louis XVI. Period. 
 Lady Warwick . . . . . . . . . . La Reine Marie Antoinette. 
 Lady Curzon  . . . . . . . . . . La Reine Marie Leszuiska. 
 Lady Kilmorey  . . . . . . . . . Madame du Barry. 
 Lady Algernon Gordon-Lennox  . . Princesse de Lamballe. 
 Lady Burton  . . . . . . . . . . Madame de Tençin. 
 Lady Garvagh . . . . . . . . . . Comtesse d'Artois. 
 The Hon. Mrs. Greville . . . . . Madame Elizabeth de France. 
 The Hon. Mrs. George Keppell . . Madame de Polignac. 
 Lady Rose Leigh  . . . . . . . . Duchesse de Villars. 
 Mrs. Farquharson . . . . . . . . L'Archiduchesse Louise. 
 Miss Naylor  . . . . . . . . . . Comtesse de Charny. 
 The Hon. Mrs. Sackville West . . Duchess of Dorset. 
 Lord Chelsea . . . . . . . . . . Le Roi Louis XV. 
 Lord Camden  . . . . . . . . . . Duc de Richelieu. 
 The Hon. Humphrey Sturt  . . . . Abbé de l'Epoque. 
 Lord Burton  . . . . . . . . . . Cardinal Dubois. 
 Lord Annaly  . . . . . . . . . . )
 Lord Tullibardine  . . . . . . . )
 Lord George Murray . . . . . . . )
 The Hon. F. Guest  . . . . . . . )
 Sir Samel Scott  . . . . . . . . ) Mousquetaires et Mili- 
 Captain Gordon Wilson  . . . . . ) taires de l'Epoque. 
 Captain the Hon. W. Lambton  . . )
 Captain Gilbert Elliot . . . . . )
 Mr. Dugdale  . . . . . . . . . . )
 Mr. Clive Wilson . . . . . . . . Le Comte de Ferson. 
 Mr. W. M. Lowe . . . . . . . . . Gentilhomme de la Cour Louis XVI. 
 Mr. Arnold Morley  . . . . . . . Gentilhomme de la Cour Louis XV.&lt;ref name=&quot;:0&quot; /&gt;{{rp|p. 7, Col. 6B}}

===Elizabethan===
Elizabethan costumes seem to have been very popular, even outside the Elizabethan procession, in part because the Royals were costumed in Renaissance styles as well. In this description the ''Gentlewoman'' includes Anne of Austria and the Electress of Luneberg and Hanover, not to mention Napoleon, who would not have been in Elizabeth's court. They could conceivably have walked in the Elizabeth procession, although Anne of Austria, while Renaissance, might have walked with the French procession if it included Louis XIII. The Duchess of Connaught and the Duchess of Teck were very closely related to Queen Victoria's immediate family and thus might be grouped here because many of the Royals wore Elizabethan dress.

&lt;blockquote&gt;A Court, the details of which were perfectly carried out, was that of Elizabeth of England. Lady Tweedmouth took the part of Her Majesty, and her costume was an exact reproduction of Queen Elizabeth's portrait in the National Portrait Gallery. Her skirt of rich old white and gold brocade was held in place by the old-fashioned hoops, the bodice and front of gold tissue embroidered in old jewels were finished by stiffened cuffs and large wired collar of old lace wrought with gold. Four yeomen held a canopy over Her Majesty's head. Their uniforms were exactly copied from the picture of the Field of the Cloth of Gold at Hampton Court. These were the Duke of Roxburghe, the Hon. Dudley Marjoribanks, Captain Maunde Thompson, and Mr. Rose attired in scarlet and black. The two heralds who preceded the Queen were Mr. Harold Brassey and Mr. E. Villiers, while Lord Rothschild, in a splendid costume of the time, walked between them.

Among her Majesty's Court were Sir Walter Raleigh (Mr. Ernest Beckett), Sir Philip Sidney (Mr. H. Warrender), Sir Francis Drake (Sir Charles Hall), the Lord Chief Justice (Sir Francis Jeune), the Lord of Burleigh (the Earl of Sandwich), while Lord Lonsdale, who carried a hooded falcon on his wrist, represented Sir Richard Lowther. The Duchess of Roxburghe, and the Countess of Powis, as the Countess of Shrewsbury and Lady Herbert of Cherbury, looked very effective. Countess Spencer, Mrs. Habington, Lady de Ramsay, and the Archbishop of Canterbury, represented by Lord Rowton, made up the Court. Then followed Mary Queen of Scots, in the person of Lady Edmonstone, wonderfully attired in turquoise-blue velvet with pearls and white satin; Mary Hamilton, in white satin and gold, and Mary Seaton, in white, followed in her wake as did the Countess of Lonsdale (Lady Hunsdon), Lord Glenesk, and many others.

H.R.H. the Duchess of Connaught, as Anne of Austria, and H.R.H. the Duchess of Teck, as Electress of Luneberg and Hanover, looked their characters very well, and a very effective trio was formed by the Countess Clary d'Aldringen, Countess Isabel Deym, and Countess Kinsky, as the three sisters of Napoleon. The Duchess of Somerset as Lady Jane Seymour, after a picture by Holbein, was dressed in gold brocade with a wonderful headdress; superbly jewelled, white gloves and Holbein ornaments embroidered on her gown. Margaret of Orleans, impersonated by the Duchess of Manchester, in white satin and silver, was a great success. Josephine, the wife of Napoleon, copied from the picture of her coronation, was impersonated by the Marchioness of Tweeddale, who wore white satin wrought with gold, and a train of geranium-red velvet, trimmed with ermine. Lady Lathom as Catherine of Arragon was splendidly dressed in bronze-green velvet worked in gold designs.&lt;ref name=&quot;:8&quot; /&gt;{{rp|p. 32, Col. 3c – 34, Col. 1a}}&lt;/blockquote&gt;

''Truth'' describes the procession like this:&lt;blockquote&gt;Lady Tweedmouth was gorgeously arrayed as Queen Elizabeth, and was surrounded by a numerous Court, including Lord Tweedmouth, Lord Battersea, the Earl of Sandwich, and Lord Frederick Hamilton, to say nothing of six stalwart halberdiers, one of whom was the Duke of Roxburghe, whose Duchess was also bravely attired as an Elizabethan lady of high degree.&lt;ref name=&quot;:10&quot; /&gt;{{rp|41, Col 2c}}&lt;/blockquote&gt;The Elizabethan procession was &quot;led&quot; by Fanny Marjoribanks, [[Social Victorians/People/Tweedmouth|Lady Tweedmouth]] (at 85) as Queen Elizabeth, but she did not come first in the procession.

# Heralds
# Row of 3 men
#* Mr. E. ([[Social Victorians/People/Grimthorpe|Ernest) Beckett]] (at 313), as Sir Walter Raleigh
#* Mr. H. (Hugh) Warrender (at 314), as Sir Philip Sydney
#* [[Social Victorians/People/Charles Hall|Sir Charles Hall]], Q.C., M.P. (at 127), as Sir Francis Drake at the head of the Queen Elizabeth procession
# Row of 3 men
#* Sir F. ([[Social Victorians/People/Jeune|Francis) Jeune]] (at 315), as Lord Chief Justice
#* Edward Montagu, [[Social Victorians/People/Sandwich|8th Earl of Sandwich]] (at 71), as Lord Burleigh
#* [[Social Victorians/People/Lowther|Earl of Lonsdale]], Sir Richard Lowther (at 225), as Lord High Falconer
# Row of 3 women
#* Violet [[Social Victorians/People/Powis|Countess Powis]] (at 316), as Lady Herbert of Cherbury in the Queen Elizabeth procession
#*Anne [[Social Victorians/People/Roxburghe|Duchess of Roxburghe]] (at 22), as Countess of Shrewsbury or Bess of Hardwicke
#*Charlotte, [[Social Victorians/People/Spencer|Countess Spencer]] (at 192), as the Countess of Lennox
# Row of 2 women
#* Grace, [[Social Victorians/People/Lowther|Countess Lonsdale]] (at 211), as the Countess of Essex
#*Mrs. [[Social Victorians/People/Cavendish Bentinck#Mrs. and Mr. Arthur James|Mary (Arthur) James]] (at 318), as Elisabeth Cavendish
# [[Social Victorians/People/Leslie|Colonel John Leslie]] (at 261) was dressed as &quot;Lord Darnley (carrying Sword of State)&quot; in the Queen Elizabeth procession&lt;ref&gt;&quot;Leslie as Earl Darnley.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158503/Sir-John-Leslie-2nd-Bt-as-Earl-Darnley.&lt;/ref&gt;
# Row of 2 men
#* Edward Marjoribanks, [[Social Victorians/People/Tweedmouth|Lord Tweedmouth]] (at 109) as the Earl of Leicester
#* George Capell, 7th [[Social Victorians/People/Essex|Earl Essex]] (at 64), as his ancestor, Robert Devereux, 2nd Earl of Essex
* Yeomen
** The [[Social Victorians/People/Roxburghe|Duke of Roxburghe]] (at 49), a halberdier attending on Queen Elizabeth
** Dudley Marjoribanks (at 319), son of [[Social Victorians/People/Tweedmouth|Lord and Lady Tweedmouth]], as a Yeoman bearing the canopy, possibly, with the Duke of Roxburghe and two &quot;brother officers in the Royal Horse Guards&quot;
* Row before the queen, with canopy
** [[Social Victorians/People/Ephrussi|Mr. Ephrussi]] (at 320), as the Spanish Envoy
** Canopy
** H. E. M. [[Social Victorians/People/de Courcel|Alphonse Chodron de Courcel]] (at 133), as the French Ambassador
* Fanny Marjoribanks, [[Social Victorians/People/Tweedmouth|Lady Tweedmouth]] (at 85), as Queen Elizabeth
*More Yeomen (possibly the two &quot;brother officers in the Royal Horse Guards&quot; of Dudley Marjoribanks (at 319)
**[[Social Victorians/People/Mann Thomson|Captain Mann Thomson]] (at 321), as a Yeoman
**[[Social Victorians/People/Rose|Mr. Rose]] (at 322), as a Yeoman
*Row behind the queen
**[[Social Victorians/People/Edmonstone|Sir A. Edmonstone]] (at 323), as Duc d'Alencon
**[[Social Victorians/People/Holden|Henry Holden]] (at 325), as Will Somers (Court Jester)
**John Spencer, 5th [[Social Victorians/People/Spencer|Earl Spencer]] (at 145), as Sir A. Brown, First Viscount Montagu (None of the sources agree on who he personated; this is the ''Morning Post'')
*More Yeomen (possibly the two &quot;brother officers in the Royal Horse Guards&quot; of Dudley Marjoribanks (at 319)
** [[Social Victorians/People/Villiers Schott|Mr. E. Villiers]] (at 326), as a Yeoman
**[[Social Victorians/People/Brassey|Harold Brassey]] (at 253), as a Yeoman
* Row behind the yeomen
** Arthur [[Social Victorians/People/Arran|Earl of Arran]] (at 327), as Cardinal Loraine
** Nathan Mayer de Rothschild, [[Social Victorians/People/Rothschild Family|Lord Rothschild]] (at 216), as Swiss Burgher
** Montagu Lowry-Corry, [[Social Victorians/People/Rowton|1st Baron Rowton]] (at 189), as Archbishop Parker, the Archbishop of Canterbury
*Row of 3 women
**Margaret (the [[Social Victorians/People/Greville|Hon. Mrs. Ronald) Greville]] (at 298), as Mary Seaton
**[[Social Victorians/People/Edmonstone|Lady Edmonstone]] (at 324), as Mary Queen of Scots
**Mary Louise Douglas-Hamilton, [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Duchess of Hamilton]] (at 166), as Mary of Hamilton

*Row of 2 women
**Constance de Rothschild Flower, [[Social Victorians/People/Rothschild Family|Lady Battersea]] (at 328), as Lady Hunsdon
**Rosamond Fellowes, [[Social Victorians/People/de Ramsey|Lady de Ramsey]] (at 329), as Lady Burleigh
*Row of 3 men
**[[Social Victorians/People/Rothschild Family|Baron Ferdinand de Rothschild]] (at 330), as Casimir Count Patatine of Bavaria, an Austrian noble of the 16th century
**George, [[Social Victorians/People/Powis|Earl of Powis]] (at 317), as Lord Herbert of Cherbury
**Cyril Flower, 1st [[Social Victorians/People/Rothschild Family|Baron Battersea]] (at 110) as Lord Hunsdon
*Row of 3 men
**Mr. [[Social Victorians/People/Webb|Godfrey Webb]] (at 331), as Martin Frobisher
**Algernon [[Social Victorians/People/Borthwick|Borthwick, Baron Glenesk]] (at 87) as Lord James Murray
**The Hon. S. (George William [[Social Victorians/People/Lyttelton|Spencer) Lyttelton]] (at 332), as Sir Thomas Gresham
*[[Social Victorians/People/Peel Family|Rochfort Maguire]] (at 241) was C. Maguire, Lord of Fermanagh
The royals were mostly dressed in Elizabethan dress but are properly considered their own group. Besides them, these people were in Elizabethan dress at the ball but not listed as being in the Elizabeth procession or quadrille:
*[[Social Victorians/People/Abercorn|Lord Frederick Hamilton]] (at 84), a Gentleman of the Court of Queen Elizabeth
*[[Social Victorians/People/Abercorn|Mr. Ronald Hamilton]] (at 105), a Gentleman of the Court of Queen Elizabeth
* Walter, [[Social Victorians/People/Mar and Kellie|Earl of Mar and Kellie]] (at 58), as an Elizabethan or perhaps as Sir Walter Scott
* Henry Lascelles, 5th [[Social Victorians/People/Harewood|Earl Harewood]] (at 62), as Philip II of Spain
*[[Social Victorians/People/Crewe-Milnes|Robert Offley Ashburton Crewe-Milnes]] (at 169), also as Philip II of Spain
* [[Social Victorians/People/Jarvis|Weston (Mr. A. W.) Jarvis]] (at 106), as Sir Francis Walsingham
* Emma Louise von Rothschild, [[Social Victorians/People/Rothschild Family|Lady Rothschild]] (at 112), as Anne of Cleves
* Lady Alice Villiers Bootle-Wilbraham, the [[Social Victorians/People/Lathom|Countess of Lathom]] (at 213), as Catharine of Aragon
*[[Social Victorians/People/Prince Charles of Denmark|Prince Charles of Denmark]] accompanied the Prince and Princess of Wales as a gentleman of the Court of Denmark in the time of Elizabeth
*Susan St. Maur, [[Social Victorians/People/Somerset|Duchess of Somerset]] (at 209), as Jane Seymour

*[[Social Victorians/People/Katharine Mary Montagu Douglas Scott|Lady Katharine Montagu-Douglas-Scott]] (at 25), as Mary Queen of Scots; perhaps she walked in this procession.
* Algernon St. Maur, [[Social Victorians/People/Somerset|Duke of Somerset]] (at 27), as Somerset the Protector, older brother of Jane Seymour. (The ''Gentlewoman'' lists the Duchess of Somerset as a member of the Queen Elizabeth procession, but the ''Morning Post'' and the ''Times'' do not '''double check this'''.)
Georgiana, [[Social Victorians/People/Hindlip|Baroness Hindlip]] was supposed to be part of the Elizabethan procession, but she and Samuel, [[Social Victorians/People/Hindlip|Baron Hindlip]] did not attend, as he was quite ill and in fact died less than two weeks later.

The ''Morning Post'' typeset a visualization of the procession, more or less, like this:
                                QUEEN ELIZABETH PROCESSION.
                                       ''Heralds''. 
 ''Sir Walter Raleigh'',                 ''Sir P. Sydney'',                ''Sir F. Drake'',
 Mr. E. Beckett.                    Mr. H. Warrender.               Sir C. Hall.
 ''Lord Chief Justice'',                 ''Lord Burleigh'',           ''Sir Richard Lowther''
 Sir F. Jeune.                     Earl of Sandwich.         ''(Lord High Falconer)'',
                                                                 Earl of Lonsdale.
                                                          
 ''Lady Herbert of Cherbury,       Countess of Shrewsbury,       Countess of Lennox,''
 Countess of Powis.               Duchess of Roxburghe.          Countess Spencer.
      ''Countess of Essex,                                      Elisabeth Cavendish.''
     Countess of Lonsdale.                                       Mrs. A. James. [p. 7, Col. 5–6]
 
                                 ''Lord Darnley''
                           ''(carrying Sword of State),''
                                 Colonel Leslie.
 
 ''Lord Leicester,                                                    Earl of Essex,''
 Lord Tweedmouth.                                                   Earl of Essex.
              ''Yeoman,                                    Yeoman,''
        Duke of Roxburghe.                        Hon. D. Marjoribanks.
 
       ''Spanish Envoy,''               &lt;u&gt;| CANOPY. |&lt;/u&gt;                  ''French Ambassador,''
       Mr. Ephrussi.                                           H. E. M. de Courcel.
                                  ''Queen Elizabeth,''
                                  Lady Tweedmouth.
              ''Yeoman,                                    Yeoman,''
        Captain Mann Thomson.                            Mr. Rose.
 ''Duc d'Alençon,                     Will Somers                 Sir A. Brown, First''
 Sir A. Edmonstone.                ''(Court Jester).                 Viscount Montagu.''
                                    Mr. Holden.                      Earl Spencer.
 
              ''Yeoman,                                    Yeoman,''
          Mr. E. Villiers.                          Mr. Harold Brassey.
 ''Cardinal Loraine,                 Swiss Burgher,          Archbishop of Canterbury.''
   Earl of Arran.                 Lord Rothschild.                      Lord Rowton.
 
  ''Mary Seaton,                   Mary Queen of Scots,                  Mary Hamilton'',
 Hon. Mrs. Greville.                 Lady Edmonstone.            Duchess of Hamilton.
           ''Lady Hunsdon,                                 Lady Burleigh,''
          Lady Battersea.                               Lady de Ramsey.
 ''Casimir Count Patatine of Bavaria  Lord Herbert of Cherbury,            Lord Hunsdon,''
 Baron F. de Rothschild.                Earl of Powis.                 Lord Battersea.
 ''Martin Frobisher,               Lord James Murray,                Sir Thomas Gresham,''
 Mr. Godfrey Webb.                   Lord Glenesk.                  Hon. S. Lyttelton.
                              ''C. Maguire (Lord of Fermanagh),''
                                    Mr. R. Maguire.&lt;ref name=&quot;:0&quot; /&gt;{{rp|7, Col. 5C–6B}}

==Other Groups==
Other processions or quadrilles existed, not captured fully by the ''Morning Post'' but mentioned in, for example, the London ''Daily News'' story about the ball.The ''Morning Post'' is not complete in other ways as well, judging by other newspaper accounts or even descriptions from later in the big ''Morning Post'' story. Both the Knights of the Round Table of King Arthur procession and the Cosway Quadrille are examples of processions or quadrilles not detailed in the ''Morning Post''.

=== The Court of Marguerite de Valois ===
Theoretically, the court of Marguerite de Valois could have been included among the Elizabeth procession, but some of the people in this court, which was led by the Alexandra, Princess of Wales, might have been on the dais with her, so perhaps it could be imagined as a court but not a procession.
* [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales]], as Marguerite Valois
*[[Social Victorians/People/Alington|Evelyn, Lady Alington]] (at 41), as Duchesse de Nevers
* Princess Charles of Denmark (Princess Maud of Wales)
* Prince Charles of Denmark was in this court? He was photographed with Princess Charles of Denmark and Princess Victoria of Wales; this photograph is in the Album.&lt;ref name=&quot;:6&quot;&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.&lt;/ref&gt;
* [[Social Victorians/People/George and Mary|Mary Teck, Duchess of York]], attended by
** [[Social Victorians/People/Beauchamp|Lady Mary Lygon]] (at 547), as Marie de Lorraine, a lady of the Court of Marguerite de Valois
** Sir Charles Cust (at 152)
* Princess Louise, [[Social Victorians/People/Fife|Duchess of Fife]] (at 177)
* [[Social Victorians/People/Princess Victoria of Wales|Princess Victoria of Wales]] (at 370)
* Princess Victoria of Schleswig-Holstein (at 10)

===Queen Guinevere and the Knights of the Round Table of King Arthur procession===
A procession represented King Arthur's Round Table. This court is mentioned in the story in the ''Times'' as well as the Evening ''Mail'', which seems to have reprinted the story from the ''Times''.&lt;ref name=&quot;:5&quot; /&gt;&lt;ref name=&quot;:9&quot; /&gt;{{rp|p. 9, Col. 1c}} The ladies included Lady Ormonde, Lady Constance Butler, Lady Ashburton and Miss Chaplin. 

According to the ''Morning Post'' and the ''Gentlewoman'', the Knights of the Round Table were George, [[Social Victorians/People/Rodney|Baron Rodney]]; [[Social Victorians/People/Grosvenor|Hon. R. Grosvenor]]; Seymour Henry Bathurst, [[Social Victorians/People/Bathurst|7th Earl Bathurst]]; and Hon. Grosvenor [[Social Victorians/People/Hood|Hood]].&lt;ref name=&quot;:0&quot; /&gt;{{rp|p. 8, Col. 1b}} &lt;ref name=&quot;:8&quot; /&gt;{{rp|p. 40, Col. 1c}} According to the ''Daily News,'' the Knights of the Table Round were &quot;[[Social Victorians/People/Ashburton|Lord Ashburton]], Lord Rodney, [[Social Victorians/People/Bathurst|Lord Bathurst]], [[Social Victorians/People/Ampthill#Oliver Russell, Lord Ampthill|Lord Ampthill]], and [[Social Victorians/People/Beauchamp|Lord Beauchamp]]&quot;;&lt;ref name=&quot;:1&quot; /&gt;{{rp|p. 5, Col. 7a}} the newspaper accounts disagree on Lord Beauchamp in particular. George, Baron Rodney was 40 years old at the time of the ball; Seymour Henry Bathurst, [[Social Victorians/People/Bathurst|7th Earl Bathurst]] was nearly 33; Hon. Grosvenor [[Social Victorians/People/Hood|Hood]] was 29;  [[Social Victorians/People/Ashburton|Lord Francis Ashburton]] was nearly 31; [[Social Victorians/People/Ampthill|Lord Ampthill]] was 28; [[Social Victorians/People/Beauchamp|Lord Beauchamp]] was 25. We can see what they wore because some of them had their portraits taken in their costume.
# Elizabeth [[Social Victorians/People/Ormonde|Butler]], the [[Social Victorians/People/Ormonde|Marchioness of Ormonde]] (at 373), was dressed as Guinevere
#Lord [[Social Victorians/People/Westminster#Lord Gerald Grosvenor|Gerald Grosvenor]] (at 618), as Sir Launcelot  (listed in the ''Times'' and the Evening ''Mail'', not in the ''Morning Post'')
#Lord [[Social Victorians/People/Westminster#Lord Arthur Grosvenor|Arthur Grosvenor]] (at 619), Arthur Hugh Grosvenor, as King Arthur (listed in the ''Times'', not in the ''Morning Post'')
# [[Social Victorians/People/Rodney|Corisande, Baroness Rodney]] (at 472) was also dressed as Queen Guinevere, according to her portrait in the Album in the National Portrait Gallery,&lt;ref&gt;&quot;Corisande Evelyn Vere.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158471/Corisande-Evelyn-Vere-ne-Guest-Lady-Rodney-as-Queen-Guinevere.&lt;/ref&gt; but she may not have been in this procession, although George, Baron Rodney was, as a Knight of the Round Table.
# [[Social Victorians/People/Ormonde|Lady Constance Butler]] (Elizabeth Butler's daughter, at 374) was Lynette or Elaine
# Mabel, [[Social Victorians/People/Ashburton|Lady Ashburton]] (at 375), as Enid
# [[Social Victorians/People/Henry Chaplin|Miss Chaplin]] (probably Hon. Edith Helen Chaplin, at 407), as Elaine
#Mr. [[Social Victorians/People/Henry Chaplin|Eric Chaplin]] (at 616), as Sir Gareth
# John [[Social Victorians/People/Lister-Kaye|Lister Kaye]] (at 97), as Sir Kay
#[[Social Victorians/People/Tilney|Mr. Tilney]] (at 615), as Sir Galahad
# [[Social Victorians/People/Peel Family|Captain R. Peel]] (at 614), as Sir Bedivere
# [[Social Victorians/People/Ampthill#Margaret Russell, Lady Ampthill|Margaret Russell, Lady Ampthill]] (at 419), as a Lady in Waiting at the Court of King Arthur
#Mr. J. B. ([[Social Victorians/People/Leigh|John Blundell) Leigh]] (at 602), as Sir Tristram
#Captain [[Social Victorians/People/Milner|George Francis Milner]] (at 617), as Sir Percevale [sic]
# Knights of the Round Table (first four, ''Morning Post''; first two plus last three are London ''Daily News'')
##[[Social Victorians/People/Rodney|Lord Rodney: George, Baron Rodney]] (at 80)
##[[Social Victorians/People/Bathurst|Earl Bathurst]] (at 82)
##[[Social Victorians/People/Grosvenor|Hon. R. Grosvenor]] (at 81)
## [[Social Victorians/People/Hood|Hon. G. Hood]] (at 83). The printing on the portrait that was in the Album presented to the Duchess of Devonshire says, &quot;The Hon. Grosvenor Hood as Sir Galahad.&quot;&lt;ref&gt;&quot;Grosvenor Hood as Sir Galahad.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158513/Grosvenor-Arthur-Alexander-Hood-5th-Viscount-Hood-as-Sir-Galahad.&lt;/ref&gt;
## Francis, [[Social Victorians/People/Ashburton|Lord Ashburton]] (at 376)
##[[Social Victorians/People/Ampthill|Oliver Russell, Baron Ampthill]] (at 77)
## William Lygon, 7th [[Social Victorians/People/Beauchamp|Earl Beauchamp]] (at 60)

===The Cosway Quadrille===
The &quot;Cosway quadrille,&quot; with the [[Social Victorians/People/Roxburghe|Ladies Innes-Ker]] and the [[Social Victorians/People/Villiers|Ladies Villiers]], is not mentioned in the ''Morning Post'' article. This description from the London ''Daily News'' suggests that there were two Ladies Ker and two Ladies Villiers: &quot;Very artistic was the &quot;Cosway&quot; quadrille, in which the Ladies Ker and the Ladies Villiers took part. The long clinging gowns of Oriental cream satin were veiled in pink muslin, and had very short waists and coloured silk sashes — two of blue and two of pink.&quot;&lt;ref name=&quot;:1&quot; /&gt;{{rp|6, Col. 1a}} These costumes seem to have been based on portraits by Richard rather than Maria Cosway. The Ladies Innes-Ker had the blue and the Ladies Villiers had the pink sashes.

[[Social Victorians/People/Roxburghe|Lady Margaret Innes-Ker]] (at 23) and [[Social Victorians/People/Roxburghe|Lady Victoria Innes-Ker]] (at 383) are in the album given to the Duke and Duchess of Devonshire by some of the people attending the ball.&lt;ref name=&quot;:6&quot; /&gt; [[Social Victorians/People/Villiers|Lady Edith Villiers]] (at 282) was dressed after Cosway and may have been in the Quadrille. The other Lady Villiers is not likely to be [[Social Victorians/People/Jersey|Lady Margaret Childs-Villiers]] (at 433), called Lady M. Villiers. We know Lady Margaret Villiers was already at the ball: her portrait as Madame Henriette Duchess d'Orleans is in the Album. She is not part of the same family Lady Edith came from, which was that of the [[Social Victorians/People/Villiers|Earl of Clarendon]]; Lady Margaret Childs-Villiers is part of the family of the [[Social Victorians/People/Jersey|Earl of Jersey]]. Another young woman from that family was [[Social Victorians/People/Jersey|Lady May Julia Child-Villiers]], who may in fact be the second of the Ladies Villiers in the Quadrille.

Since a quadrille is usually a dance for four couples, this list would make up the Cosway quadrille if indeed four women took part:

# [[Social Victorians/People/Roxburghe|Lady Margaret Innes-Ker]] (at 23)
# [[Social Victorians/People/Roxburghe|Lady Victoria Innes-Ker]] (at 383)
# [[Social Victorians/People/Villiers|Lady Edith Villiers]] (at 282)
# [[Social Victorians/People/Jersey|Lady May Julia Childs-Villiers]] (at 372)
Two other women were dressed after Cosway, neither from any of the Villiers families. Both Miss [[Social Victorians/People/Stanley#Madeline%20Stanley|Madeline Stanley]] (at 552) and [[Social Victorians/People/Marion Margaret Violet Lindsay Manners|Violet Manners, Marchioness of Granby]] (at 448) were in the Album. Miss Stanley was dressed as Lady Eliza Hopeton, &quot;after a miniature by Cosway,&quot; and Lady Violet was dressed &quot;after Cosway&quot; as well.

==== People Whose Costumes Were &quot;After Cosway&quot; ====

* 2 Ladies Ker (if the London Daily News article is right)
** [[Social Victorians/People/Roxburghe|Lady Margaret Innes-Ker]] (at 23)
** [[Social Victorians/People/Roxburghe|Lady Victoria Innes-Ker]] (at 383)
* 2 Ladies Villiers (if the London ''Daily News'' article is right)
** L[[Social Victorians/People/Villiers|ady Edith Villiers]] (at 282)
** [[Social Victorians/People/Jersey|Lady May Julia Child-Villiers]] (at 372)?
** [[Social Victorians/People/Jersey|Lady Margaret Childs-Villiers]] (at 433), as Madame Henriette Duchess d'Orleans
* [[Social Victorians/People/Marion Margaret Violet Lindsay Manners|Violet Manners, Marchioness of Granby]] (at 448)
* Miss [[Social Victorians/People/Stanley#Madeline Stanley|Madeline Cecilia Carlyle Stanley]] (at 552)

=== Courts of Charles I (and Henrietta Maria, Queen) and Charles II ===
A number of people were identified as being a member of the courts of Charles I or Charles II, even though no such group is directly discussed in the major stories. This is Charles II of England; the Duke of Devonshire was dressed as Charles V, Holy Roman Emperor of Germany, so this was not a group forming around him. Relevant people would be Cromwell, the Roundheads, and so on.

#[[Social Victorians/People/Zetland|Lilian, Marchioness of Zetland]] (at 48), Henrietta Maria, wife of Charles I, after Van Dyck
#[[Social Victorians/People/Manchester|Lord Charles Montagu]] (at 161), as Charles I, after Van Dyck
#[[Social Victorians/People/Harcourt#Elizabeth Harcourt|Lady Elizabeth Harcourt]] (at 94), as a Lady of the Court of Henrietta Maria
# Lord Edward Cecil (Edward Herbert [[Social Victorians/People/Salisbury|Gascoyne-Cecil]]) (at 411), as a courtier of Charles I
#[[Social Victorians/People/Chaine|Colonel William Chaine]] (at 98), as a Gentleman of the Court of Charles II
# Mr. [[Social Victorians/People/Portland|Cavendish-Bentinck]] (at 113), as a Gentleman of the Court of Charles II
# Sir [[Social Victorians/People/Donald Mackenzie Wallace|Donald Mackenzie Wallace]] (at 114), as a Gentleman of the Court of Charles II
# Sir Henry [[Social Victorians/People/Meysey-Thompson|Meysey-Thompson]] (at 116), in the Costume of a gentleman of the period of Charles II
# Herbert Gardner, [[Social Victorians/People/Burghclere|Lord Burghclere]] (at 129) as a Puritan
# William, [[Social Victorians/People/Portland|Duke of Portland]] (at 28), as &quot;Steenie&quot; Villiers, 1st Duke of Buckingham, according to the London ''Daily News''&lt;ref name=&quot;:1&quot; /&gt; and the ''Pall Mall Gazette''&lt;ref name=&quot;:7&quot;&gt;“The Devonshire House Ball. A Brilliant Gathering.” The ''Pall Mall Gazette'' 3 July 1897, Saturday: 7 [of 10], Col. 2a–3a. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18970703/019/0007.&lt;/ref&gt;; the ''Morning Post''&lt;ref name=&quot;:0&quot; /&gt; and the ''Times''&lt;ref name=&quot;:5&quot; /&gt; say he went as Duca Filiberto di Savoia in the 17th-century section of the Venetians procession, where he is listed as well.
# [[Social Victorians/People/Portland#George Cavendish-Bentinck|George (William George) Cavendish-Bentinck]] (at 666), as William, Baron Bentinck, A.D. 1643
# [[Social Victorians/People/Zetland|Lawrence Dundas, Marquis of Zetland]] (at 59), as the Duke of Buckingham (probably at the time of Charles I or II)

== People Not Listed as Part of a Procession or Quadrille ==

=== Subnetworks ===
'''Probably more than half''' the people who came in costume were not part of an organized procession or quadrille. Made up largely of courts of monarchs, and particularly women who were monarchs or leaders, the processions have a kind of internal coherence as people attending this ball were able to find people from those courts whom they were willing and able appear as. Because it would have taken communication and negotiation for people to determine and claim their place in the courts, the processions and quadrilles suggest that they make up subnetworks of people within the larger network of those who attended. The quadrilles would have been expected to rehearse, and we know that at least the Queen Elizabeth court met the night before the ball for dinner.

Newspaper articles about the ball, the people who attended, and the costumes they wore reveal how this social world was dominated and organized by women's identities and practices. Most obviously, the party is called the Duchess of Devonshire's ball by everyone, then and now. The highest-status women present were queens or princesses in history, biblical stories, and legends, from Louisa, the Duchess of Devonshire's Zenobia to Queen Elizabeth, Empress Marie-Thérèse, and Catherine II of Russia. The courts were organized around these women. Leonore Davidoff discusses some of the implications of women's roles as gatekeepers in this social world at this time in her ''The Best Circles: Society Etiquette and the Season''.&lt;ref name=&quot;:11&quot;&gt;Davidoff, Leonore. The ''Best Circles: Society Etiquette and the Season''. Intro., Victoria Glendinning. The Cressett Library (Century Hutchinson), 1986 (orig ed. 1973).&lt;/ref&gt;

The fact that the courts were led by women is made clear in the newspaper reportage, sometimes in spite of overt language to the contrary. For example, what the newspapers call the courts of '''Louis XIV???''' are in fact the court of Marie Antoinette, led by Daisy, Countess of Warwick.

Outside of this kind of organized coherence, of whatever degree, small groups of people decided to dress to reflect relationships, most often husbands and wives.

Inevitably, working independently from each other, more than one person went to the ball dressed as the same person from history or even the same character from a novel, opera, or play. A few were men, but most were women, or at least most of those reported were women. There were multiple

* Cleopatras
** [[Social Victorians/People/Pless#Daisy, Princess Henry of Pless|Daisy, Princess Henry of Pless]] (at 38), or as the Queen of Sheba
** [[Social Victorians/People/Paget Family#Minnie Paget|Minnie Paget]], Mrs. Arthur Paget (at 90)
* Princesses de Lamballe
** Countess [[Social Victorians/People/Deym#Countess Isabel Deym|Isabel Deym]] (at 67)
** [[Social Victorians/People/John Milton Hay#Mr. John and Mrs. Clara Hay|Mrs. Clara Hay]] (at 153)
** Lady [[Social Victorians/People/Gordon-Lennox#Lady Blanche Gordon-Lennox|Blanche Gordon-Lennox]] (at 333)
** [[Social Victorians/People/Ampthill#Emily, Lady Ampthill|Emily, Lady Ampthill]] (at 420)
* Queens of Sheba
* Elizabeth, Queens of Bohemia
** [[Social Victorians/People/Cadogan|Beatrix, Countess Cadogan]] (at 55), as Elizabeth, Queen of Bohemia, after a painting by Honthorst
** Lady Ethel [[Social Victorians/People/Meysey-Thompson|Meysey Thompson]] (at 391), as Elizabeth, Queen of Bohemia
** [[Social Victorians/People/Westminster|Katherine Grosvenor, Duchess of Westminster]] (at 34), as Queen Elizabeth of Bavaria (who might not be Elizabethan, but [[Social Victorians/People/Rothschild Family|Baron Ferdinand de Rothschild]], as Casimir Count Patatine of Bavaria was in the Elizabethan procession)
* Anne of Austria, Queen of France
** Anne of Austria, Queen of France ([[Social Victorians/People/Connaught|Princess Louise, Duchess of Connaught]])
** Anne of Austria, Queen of France ([[Social Victorians/People/Jersey|Margaret Child-Villiers, Countess of Jersey]])

*Titania, Queen of the Fairies
**Mademoiselle de Alealo Galiano
**Susannah Graham Menzies
*Night
**Agnes, Lady Herschell
**Marie, Baroness de Courcel
*Napoleons

Men are not absent from the reportage, of course, and sometimes, as with the Prince of Wales or men mentioned as part of a couple or when their wives were not. Certainly with the Prince of Wales, a principle other than gender is at work: he is the monarch of this social world of women, in some ways the way his mother was monarch of the political world of men.

=== Individuals and Their Costumes ===
The Royals, obviously, would not have been part of any procession or quadrille because they were on the dais instead. Others, whose costumes are described in enough detail for us to know how they were dressed are listed here.
# Hon. Oliver [[Social Victorians/People/Borthwick|Borthwick]] (at 89), dressed as Marshal Turenne (reign of Louis XIV) or an officer d'Infanterie
#[[Social Victorians/People/Paget Family|Colonel Arthur Paget]] (at 91), dressed as Edward the Black Prince
# Arthur Balfour (at 86), the [[Social Victorians/People/Balfour|Right Hon. A. J. Balfour]], in a Dutch costume of 1660
# [[Social Victorians/People/Harcourt#Sir William Harcourt|Sir William Harcourt]] (at 128) as Sir Simon Harcourt, the first Lord Harcourt, in 1712, as Lord Chamberlain
#[[Social Victorians/People/Salisbury|Lady Gwendolen Cecil]] (at 404), as Portia
# [[Social Victorians/People/Cavendish|Lady Edward Cavendish]] (at 393), as Madame de Maintenon
# Alfred [[Social Victorians/People/Rothschild Family|Rothschild]] (at 605), as a French noble of the 16th century
# Sir [[Social Victorians/People/Hartopp|Charles Hartopp]] (at 111), as Napoleon I
#Lady [[Social Victorians/People/Hartopp|Millicent Hartopp]] (at 488), as the Empress Josephine
# Mr. [[Social Victorians/People/Guest|Montague Guest]] (at 115), as Montague Bertie, second Earl of Lindsey
# Mr. [[Social Victorians/People/Arthur Stanley Wilson|Arthur Wilson]] (at 118), in a costume from a portrait by Velasquez
# Mary (Mrs. Arthur) [[Social Victorians/People/Arthur Stanley Wilson|Wilson]] (at 395) wore a dress in the Georgian period
# [[Social Victorians/People/Lathom|Lady Edith Wilbraham]] (at 119), as Peg Woffington
# Lord [[Social Victorians/People/Henry James|James of Hereford]] (at 122), as Sir Thomas More
# Miss [[Social Victorians/People/Henry James|James]] (at 396), as Eugénie Hortense de Beauharnais, Louis Bonaparte's wife
# [[Social Victorians/People/Talbot|Lord Edmund Talbot]] (at 123), as a Gentleman of the Spanish Court of the early 17th Century
# [[Social Victorians/People/Wyndham|Colonel Sir Charles Wyndham Murray]], &quot;Mr. C. [[Social Victorians/People/Wyndham|Wyndham]], M.P.,&quot; (at 124), as the Emperor John Polaeologus II on his State visit to Venice in 1438.
# Lilian Maud [[Social Victorians/People/Marlborough|Spencer-Churchill]] (at 571), as a Watteau shepherdess
# Norah Beatrice Henriette [[Social Victorians/People/Marlborough|Spencer-Churchill]] (at 572), also as a Watteau shepherdess
# John Lambton, 3rd [[Social Victorians/People/Durham#John Lambton, 3rd Earl of Durham|Earl of Durham]] (at 141), as the Duc de Nemours, period Henri III
#[[Social Victorians/People/Howe|Isabella, Countess Howe]] (at 489), as Lady Howe of 1758
#Richard George Penn Curzon, [[Social Victorians/People/Howe|Viscount Curzon]] (at 197), as Admiral Lord Howe, husband of the 1758 Lady Howe, accompanied his mother
# Captain [[Social Victorians/People/Holford|George Holford]] (at 385), as Philip IV of Spain
# Francis [[Social Victorians/People/Gathorne-Hardy|Gathorne-Hardy]] (at 352), as in either the Archduchess Marie-Karoline and Emperor Joseph II section of the Austrian Court of Maria Theresa Quadrille or as a gentleman of the Court of Louis XV
# The [[Social Victorians/People/Grand Duke Michael of Russia|Grand Duke Michael of Russia]] (at 8), as Henri IV of Navarre and France, first married to Marguerite de Valois but father to the children of Gabrielle d'Estrées, personated by [[Social Victorians/People/Grand Duke Michael of Russia|Sophia, Countess de Torby]] (at 184), his morganatic wife.
#[[Social Victorians/People/Buccleuch|Louisa Jane, Duchess of Buccleuch]] (at 24), as Elizabeth, Duchess of Buccleuch, after a painting by Sir Joshua Reynolds
#[[Social Victorians/People/Buccleuch|William, Duke of Buccleuch]] (at 20), as either William Cavendish, Duke of Newcastle or Charles I
#[[Social Victorians/People/Constance Anne Montagu Douglas Scott|Lady Constance Montagu-Douglas-Scott]] (at 26), as a Watteau shepherdess
#[[Social Victorians/People/Dudley|William, Earl of Dudley]] (at 63), as Prince Rupert, so he could have been in the Charles I or Charles II procession but is not listed in any of the newspaper reports as being in a procession or quadrille; his wife, Rachel, [[Social Victorians/People/Dudley|Countess of Dudley]] (at 31) is not the Lady Dudley in the Duchesses procession; that's probably his mother, [[Social Victorians/People/Dudley|Georgina, Dowager Countess Dudley]].
#[[Social Victorians/People/Alva|Carlos, 16th Duke of Alba]] (at 32), as his ancestor at the Court of Philip II of Spain
#[[Social Victorians/People/Derby|Constance Stanley, Countess of Derby]] (at 36), as Duchess of Orleans. (Members of her family, the Stanleys, are in the Duchesses and the Louis XV and Louis XVI court processions.
#[[Social Victorians/People/Forbes|Lady Angela St. Clair-Erskine Forbes]] (at 37), as Queen of Naples
# Lawrence Dundas, Earl of [[Social Victorians/People/Zetland|Ronaldshay]] (at 529), as Sir Peter Teazle
#[[Social Victorians/People/Warwick|Lady Eva Greville Dugdale]] (at 409), as great-aunt Lady Anne Bingham
#[[Social Victorians/People/Ellesmere|Francis Egerton, 3rd Earl of Ellesmere]] (at 68), as James I
#[[Social Victorians/People/Deym|Count Franz Deym]] (at 66), as General Wallenstein
#[[Social Victorians/People/Deym|Countess Isabel Deym]] (at 67), as the Princesse de Lamballe, one of Napoleon's sisters
#[[Social Victorians/People/Clary Aldringen|Thérèse, Countess Clary and Aldringen]] (at 191), as one of Napoleon's sisters
#[[Social Victorians/People/Kinsky|Josephine, Countess Kinsky]] (at 394), as one of Napoleon's sisters, though other Princesses de Lamballe, [[Social Victorians/People/Gordon-Lennox|Lady Blanche Gordon-Lennox]] (at 333) appeared in the Louis XV and XVI procession and [[Social Victorians/People/Ampthill#Emily, Lady Ampthill|Emily, Lady Ampthill]] (at 420), also another Princess de Lamballe
#[[Social Victorians/People/Ampthill#Emily, Lady Ampthill|Emily, Lady Ampthill]] (at 420), also as the Princess de Lamballe
#Miss [[Social Victorians/People/Ampthill#Margaret Russell, Lady Ampthill|Constance Russell]] (at 418), as a flower seller or bouquetière, period Louis XV
#[[Social Victorians/People/Selborne|Beatrix Palmer, Countess Selborne]] (at 557), as Lady Percy, after a picture by Vandyk
#[[Social Victorians/People/Selborne|William Palmer, 2nd Earl of Selborne]] (at 70), as an officer of the Duke of Marlborough's Army
#[[Social Victorians/People/Peel Family|John Seymour Wynne-Finch]] (at 680), as Cosmo, Grand Duke of Tuscany
#[[Social Victorians/People/Ilchester|Lady Muriel Fox Strangways]] (at 403), as one of Queen Charlotte's bridesmaids
#Frederick, [[Social Victorians/People/Wolverton|Baron Wolverton]] (at 79), as King Richard, Coeur de Lion
#[[Social Victorians/People/Harcourt#Lewis Harcourt|Lewis Harcourt]] (at 669), as 1st Viscount Nuneham, c. 1750
#[[Social Victorians/People/Lister-Kaye|Lady Natica Lister-Kaye]] (at 499), as Duchesse de Guise in the time of Henri III
#[[Social Victorians/People/Chaine|Maria Chaine]] (at 490) as Madame Sans Gêne, from Victorien Sardou and Émile Moreau's 1893 play ''Madame Sans Gêne''
#[[Social Victorians/People/Shrewsbury|Margaret Jane Stuart-Wortley Chetwynd-Talbot]], Lady Talbot (at 485), as a Valkyrie
#[[Social Victorians/People/Salisbury|Lady Robert Cecil]] — Eleanor Lambton Gascoyne-Cecil — (at 450), as Valentina Visconti (XV Century)
#[[Social Victorians/People/Cavendish|Hon. Victor Cavendish]] (at 121), as a Tudor or an Elizabethan ambassador, from a Holbein in the National Gallery
#[[Social Victorians/People/Cavendish|Mr. S. Cavendish]] (at 700), as Count Chotak
#Louisa Montefiore, [[Social Victorians/People/Rothschild Family|Lady de Rothschild]] (at 674), as Lady Vaux, after a picture by Holbein
#Mr. [[Social Victorians/People/Rothschild Family|Leopold de Rothschild]] (at 527), as Duc de Sully
#Mrs. Leopold ([[Social Victorians/People/Rothschild Family|Marie Perugia) Rothschild]] (at 528), as Zobeida
#[[Social Victorians/People/Rothschild Family|Alfred Rothschild]] (at 605), as King Henry III
#The [[Social Victorians/People/Long|Right Hon. W. H. Long, M.P.]], (at 117), as a cavalier from the time of Charles II, after a picture by Sir Peter Lely.
#[[Social Victorians/People/Long|Mrs. Doreen Long]] (at 484) as Urania, Goddess of Astronomy or an astronomer
#[[Social Victorians/People/Argyll|Lady Elspeth Angela Campbell]] (at 621), in white with gold wings
#Madame Marie-Elisabeth [[Social Victorians/People/de Courcel|Chodron de Courcel]] (at 182), as Night
#Mademoiselle Henriette [[Social Victorians/People/de Courcel|Chodron de Courcel]] (at 371), as a Valkyrie
#Mademoiselle [[Social Victorians/People/de Courcel|Chodron de Courcel]] (at 498), as a Valkyrie
#[[Social Victorians/People/de Soveral|M. Luis de Soveral]] (at 135), as Count d'Almada, A.D. 1640
#[[Social Victorians/People/Ripon|Frederick Oliver Robinson, Earl de Grey]] (at 656), as Admiral Coligny
#[[Social Victorians/People/Rosebery|Archibald, Earl of Rosebery]] (at 139), as Horace Walpole
#[[Social Victorians/People/Gosford|Archibald Acheson, 4th Earl of Gosford]] (at 143), as Robert de la Marck (the rest of his family was in the Duchesses procession)
#Beatrix Herbert, [[Social Victorians/People/Pembroke|Countess Pembroke]] (at 146), as Mary Sidney, Countess of Pembroke, after the picture by Marcus Gheeraedts
#Sidney Herbert, [[Social Victorians/People/Pembroke|Earl Pembroke]] (at 181), as William, 1st Earl of Pembroke after Holbein
#[[Social Victorians/People/Pembroke|Lady Beatrix Frances Gertrude Herbert]] (at 648), as Signora Bacelli after Gainsborough
#Ana, Countess [[Social Victorians/People/Casa de Valencia|Casa de Valencia]] (at 148), as Nuit d'Espagne
#[[Social Victorians/People/Henry White|Daisy (Mrs. Henry) White]] (at 151), as Morosina Morosini Dogaressa of Venice
#[[Social Victorians/People/Buckingham and Chandos|Anne (Alice Anne), Duchess of Buckingham]] (at 155), as Caterina Cornaro, Queen of Cyprus
#[[Social Victorians/People/Wilbraham Egerton of Tatton|Lord Wilbraham Egerton of Tatton]] (at 591), as the Doge Morosini
#Rt. Hon. [[Social Victorians/People/Hamilton|Sir Edward Walter Hamilton]] (at 683), as John of Gaunt
#Kathleen, [[Social Victorians/People/Falmouth|Viscountess Falmouth]] (at 471), as Madame Recamier
#Albert Count von [[Social Victorians/People/Mensdorff|Mensdorff-Pouilly-Dietrichstein]] (at 180), as Henri III, King of France
#Thomas Lister, [[Social Victorians/People/Ribblesdale|4th Baron Ribblesdale]] (at 185), as Lord Ribblesdale, after the Lawrence picture of his grandfather
#Charlotte, [[Social Victorians/People/Ribblesdale|Lady Ribblesdale]] (at 206), as Duchess of Parma
#[[Social Victorians/People/Dudley|Rachel, Countess of Dudley]] (at 31), as Queen Esther
#[[Social Victorians/People/Dunraven|Lady Aileen May Wyndham-Quin]] (at 661), as Queen Hortense
#[[Social Victorians/People/Arthur Sassoon|Arthur Sassoon]] (at 553), as Chief of the Janissaries
#Mr. R. Sassoon, probably [[Social Victorians/People/Reuben David Sassoon|Reuben David Sassoon]] (at 533), as a Persian Prince
#Miss Sassoon, probably Mozelle or Louise Judith [[Social Victorians/People/Reuben David Sassoon|Sassoon]] (at 534), as a &quot;Japanese Lady&quot;
#Mary, [[Social Victorians/People/Suffolk|Countess of Suffolk]] (at 538), as a Countess of Suffolk in 1766
#[[Social Victorians/People/Suffolk|Miss Daisy Leiter]] (at 684), in what looks to be an 18th-century dress and headdress
#[[Social Victorians/People/Asquith|H. H. Asquith]] (at 381), as a roundhead
#Lord St. Oswald, Rowland Winn, [[Social Victorians/People/Saint Oswald|2nd Baron St. Oswald]], (at 641) was dressed as an officer of the Regiment de Pondichery, 1772
#The [[Social Victorians/People/Saint Oswald|Hon. Maud Winn]] (at 642), as Madame La Motte
#[[Social Victorians/People/Von Andre|Herr Von André]] (at 386), as Benvenuto Cellini
#Lady [[Social Victorians/People/Warrender|Ethel Maud Warrender]] (at 520) as &quot;Duchesse de Lauzun, La Grande Mademoiselle&quot;
#Clara ([[Social Victorians/People/John Milton Hay|Mrs. John) Hay]] (at 153), as one of the Princesses de Lamballe
#Candida Hay, [[Social Victorians/People/Tweeddale|Marchioness of Tweeddale]] (at 399), as Josephine, wife of Napoleon, with her sons Lord Arthur Vincent Hay and Lord William George Montagu Hay bearing her train (also at 399)
#William Montagu Hay, the [[Social Victorians/People/Tweeddale|Marquis of Tweeddale]] (at 400), as Saint Bris from ''Les Huguenots''
#Lady [[Social Victorians/People/Tweeddale|Clementine Hay]] (at 629), as Valentina from ''Les Huguenots''
#Mr. [[Social Victorians/People/Brett|Reginald Balioll Brett]] (at 603), as a gentleman of France
#Mrs. [[Social Victorians/People/Brett|Eleanor Frances Brett]] (at 604), as Manon Lescaut
#[[Social Victorians/People/Dyke|Lady Emily Hart Dyke]] (at 556), as an Elizabethan lady
#[[Social Victorians/People/Dunville|Violet Dunville]] (at 650), as Edith Plantagenet
#[[Social Victorians/People/Dunville|John Dunville]] (at 649), as the Emperor Yuan of China
#[[Social Victorians/People/Cole-Hamilton|Lucy Charlewood Cole-Hamilton]] (at 652), as Amy Robsart
#[[Social Victorians/People/Cole-Hamilton|Claud George Cole-Hamilton]] (at 653), as Edmund Tressilian
#[[Social Victorians/People/Cadogan|Hon. Mrs. Cadogan]] (at 668), in Elizabethan costume
#[[Social Victorians/People/Belper|Henry, Lord Belper]] (at 512), as Gentleman at Arms, time of Charles II
#[[Social Victorians/People/Maurice Baring|Hon. Maurice Baring]] (at 678), as Marlborough
#[[Social Victorians/People/Spencer|Mr. R. (Charles Robert) Spencer]] (at 493), in Elizabethan dress
#[[Social Victorians/People/Antrim|Hon. Alexander McDonnell]] (at 676), as Mercutio
#[[Social Victorians/People/Malcolm|Mr. Ian Malcolm]] (at 692), as a Courtier, time of Henry VIII
#[[Social Victorians/People/Marlborough#Lord Churchill|Lord Churchill]] (at 611), as Columbus
#[[Social Victorians/People/Leigh|Captain Gerard Leigh]] (at 570), as a member of the Life Guard, time of Charles II
#James Hamilton, [[Social Victorians/People/Abercorn|Marquis of Hamilton]] (at 657), in the period of Charles II
#Albertha Frances Anne Hamilton Spencer-Churchill, [[Social Victorians/People/Marlborough|Marchioness of Blandford]] (at 601), as a 16th-century Abbess
#Mrs. [[Social Victorians/People/Lyttelton|Edith Lyttelton]] (at 580), as a parson's daughter, after a picture by Romney
#Georgina Cavendish Coke, [[Social Victorians/People/Leicester|Countess of Leicester]] (at 516), as a Venetian lady
#Lady [[Social Victorians/People/Walsh|Clementine Walsh]] (at 523) wore an Empire costume
#Lady [[Social Victorians/People/Brassey|Violet Brassey]] (at 531), as Juliet
#Mr. [[Social Victorians/People/Brassey|Leonard Brassey]] (at 530), as Apollo
#Lord [[Social Victorians/People/Northampton|Alwyne Frederick Compton]] (at 434), as Sir William Compton, time of Charles I
#Lady [[Social Victorians/People/Northampton|Mary Compton]] (at 435), as Mme. de Chevreuse, time of Louis XIII
#The Hon. Mrs. [[Social Victorians/People/Lowther|Gwendoline Lowther]] (at 672), as Madame de Tallion (Incroyable)
#Mr. [[Social Victorians/People/Burton|J. E. Baillie]] (at 666), in a military costume of the early part of the 19th century
#The Hon. [[Social Victorians/People/Mar and Kellie#The Hon. William Erskine|William Erskine]] (at 696), as an Incroyable
#Mr. [[Social Victorians/People/Portal|William Wyndham Portal]] (at 549) in Court dress, period Marie Thérèse, or Comte de Candale from ''Un Mariage sous Louis XV''
#Florence, [[Social Victorians/People/Portal|Lady Portal]] (at 550), as Comtesse de Candale from ''Un Mariage sous Louis XV''
#Mr. [[Social Victorians/People/Cassel|Ernest Cassel]] (at 462), as Velasquez
#Mr. [[Social Victorians/People/Beit|Alfred Beit]] (at 384), as Frederick of Nassau, period 1630
#[[Social Victorians/People/Wombwell|Stephen Frederick Wombwell]] (at 671), as George Villiers, Duke of Buckingham
#The Hon. [[Social Victorians/People/Campbell|K. Campbell]] (at 695), as Charles Edward, the Pretender
#Mr. [[Social Victorians/People/Chaine|W. R. Chaine]] (at 694), as a gentleman of the Court of Queen Elizabeth
#Mr. [[Social Victorians/People/Cavendish|R. Cavendish]], probably Rt. Hon. Lord Richard Cavendish, (at 107), in a costume of the period of Marie Thérèse
#The Hon. [[Social Victorians/People/Peel Family|W. G. Peel]] (at 679), in a 15th-century Venetian costume
#Lady [[Social Victorians/People/Leicester|Mabel Coke]] (at 644), as a woodland nymph
#The Hon. [[Social Victorians/People/Suffield|Bridget Harbord]] (at 398), as the Bride of Abydos
#Lady [[Social Victorians/People/Hamilton Temple Blackwood|Florence Blackwood]] (at 637), as Flora, Goddess of Flowers
#Lord [[Social Victorians/People/Hamilton Temple Blackwood|Terence Blackwood]] (at 638), as Captain Blackwood, Royal Navy
#Hon. [[Social Victorians/People/Maurice Baring|John Baring]] (at 675), as Henry IV
#Captain [[Social Victorians/People/Durham#Captain Hedworth Lambton|Hedworth Lambton]] (at 660), as a Roman
#Major [[Social Victorians/People/Drummond|Laurence Drummond]] (at 507), as a soldier
#Hon. [[Social Victorians/People/Curzon|George Curzon]] (at 495), as a Spanish Admiral
#Sir [[Social Victorians/People/Cust|Charles Cust]] (at 152), in a soldier's uniform
#[[Social Victorians/People/Dawson|Douglas Dawson]] (at 673), as Raoul de Nangis, Les Huguenots
#[[Social Victorians/People/Dawson|Major Vesey Dawson]] (at 521), as a soldier
#Pierre, [[Social Victorians/People/Stonor#Pierre, Marquis d'Hautpoul|Marquis d'Hautpoul]] (at 387), in a Vandyck dress
#Julia Caroline Stonor, [[Social Victorians/People/Stonor#Julia Caroline Stonor, Marquise of Hautpoul|Marquis d'Hautpoul]] (at 388), as Elsa, in ''Lohengrin''
#Hon. [[Social Victorians/People/Stonor#Hon. Harry Julian Stonor|Harry Julian Stonor]] (at 389), as Lohengrin
#[[Social Victorians/People/Adair|Mrs. Adair]] (at 390), as Egyptian Queen Nitocris
#Edward Cecil Guiness, [[Social Victorians/People/Iveagh|Lord Iveagh]] (at 382), as a Cavalier, Louis XIII. period
#Mr. [[Social Victorians/People/Cavendish Bentinck|Arthur James]] (at 480), as an English gentleman of the fifteenth century
#Mr. [[Social Victorians/People/Rothschild Family#Mr. Lewis Flower|Louis Flower]] (at 506), as a French Commissary General, First Empire
#Mr. [[Social Victorians/People/William James|William Dodge James]] (at 686), as d'Artagnan
#Lady [[Social Victorians/People/Farquhar|Emilie Farquhar]] (at 639), as Duchess de Mailly, Lady in Waiting to Queen Marie Antoinette
#Sir [[Social Victorians/People/Farquhar|Horace Brand Farquhar]] (at 380), as Count Egmont or a Dutch burgher after Rembrandt
#Colonel [[Social Victorians/People/Swaine|Charles Edward Swaine]] (at 415), as an officer, 11th Dragoons, 1742
#The Right Hon. [[Social Victorians/People/Henry Chaplin|Henry Chaplin]], M.P. (at 379), as General Lefevre, First Empire
#Jesusa Murrieta, [[Social Victorians/People/Santurce|Marquisa de Santurce]] (at 633), as the Infanta of Spain
#Mr. F. de Murrieta, possibly Don José Murrieta del Campo Mello y Urritio, [[Social Victorians/People/Santurce|Marques de Santurce]] (at 634), as Philip I. of Spain
#Miss [[Social Victorians/People/Magniac|Geraldine Magniac]] (at 640), as Dawn or the Sun
#Mrs. [[Social Victorians/People/Arthur Stanley Wilson#Mrs. Florence and Mr. Charles Henry Wilson|Charles (Florence) Wilson]] (at 413), as Guinevere
#Miss [[Social Victorians/People/Burton|Jane Thornewill]] (at 712), in a costume of the Georgian era
#Mr. [[Social Victorians/People/Cork and Orrery|W. Boyle]] (at 504), in an Elizabethan costume
#[[Social Victorians/People/Mills|Hon. Violet Mills]] (at 596), in the period of Charles II
#Mrs. [[Social Victorians/People/Hartmann|Hartmann]] (at 505), as Madame Sans-Gêne
#Mr. [[Social Victorians/People/Shaftesbury|W. W. Ashley]] (at 658), as a soldier
#Mr. [[Social Victorians/People/Crichton|Herbert Creighton]] (at 647), as Charles I
#Mary [[Social Victorians/People/Murray|Graham Murray]] (at 687), as Titania
#Mr. [[Social Victorians/People/Milner|Harry (Marcus Henry) Milner]] (at 612), as a Chasseur of the Louis XV period
#Mr. [[Social Victorians/People/Strong|Arthur Strong]] (at 613), as Voltaire at the age of 25
#Mr. [[Social Victorians/People/Longhurst|A. P. Longhurst]] (at 689), as an Egyptian runner
#[[Social Victorians/People/Yznaga|Emilia Yznaga]] (at 360), as Cydalise of the Comedie Italienne from the time of Louis XV
#Laura, [[Social Victorians/People/Gleichen|Princess Victor of Hohenlohe Langenburg]] (at 16), in a Louis Quinze costume
#[[Social Victorians/People/Gleichen|Countess Helena Gleichen]] (at 17), as Joan of Arc
#Lady [[Social Victorians/People/Southampton|Hilda Southampton]] (at 402), as Beatrice
#[[Social Victorians/People/Henry Irving|Henry Irving]] (at 414), as Cardinal Wolsey
#Mrs. [[Social Victorians/People/Bischoffsheim|Clarissa Bischoffsheim]] (at 429), as Anne of Austria
#Violet Manners, [[Social Victorians/People/Marion Margaret Violet Lindsay Manners|Marchioness of Granby]] (at 448), as Isabella Marchioness of Granby
#[[Social Victorians/People/Neumann|Ludwig Neumann]] (at 452), as Le Duc de Joyeuse
#Margaret Montagu-Douglas-Scott, [[Social Victorians/People/Dalkeith|Countess of Dalkeith]] (at 460), as Helen, Countess of Dalkeith
#Lord [[Social Victorians/People/Herschell|Farrer Herschell]] (at 496), as Lord Chief Justice Sir Edward Coke
#Lady [[Social Victorians/People/Herschell|Agnes Herschell]] (at 497), as Night
#The [[Social Victorians/People/Fitzwilliam#Mrs. Edith Fitzwilliam|Hon. Edith Wentworth Fitzwilliam]] (at 635), in a costume based on a painting by Romney
#The [[Social Victorians/People/Fitzwilliam#The Hon. Reginald Fitzwilliam|Hon. Reginald Fitzwilliam]] (at 636), as Nelson
#Maud Fitzwilliam, [[Social Victorians/People/Fitzwilliam#Lord and Lady Milton|Viscountess Milton]] (at 501), as Madame Le Brun
#Catherine (the [[Social Victorians/People/Grosvenor#Hon. Catherine and Mr. Algernon Grosvenor#Hon. Catherine and Mr. Algernon Grosvenor|Hon. Mrs. Algernon) Grosvenor]] (at 510), as Marie Louise
#Hardinge Stanley Giffard, [[Social Victorians/People/Halsbury|Lord Halsbury]] (at 147), as George III
#Sir [[Social Victorians/People/Poynter|Edward Poynter]] (at 546), as Titian
#Hon. [[Social Victorians/People/Dupplin#Hon. Marie Hay|Marie Hay-Drummond]] (at 682), as Mademoiselle Andrée de Taverney, A.D. 1775
#Mrs. [[Social Victorians/People/Walker|Sophie Walker]] (at 584), as Vivien
#Mr. [[Social Victorians/People/Walker|Hall Walker]] (at 583), as Merlin
#Sir [[Social Victorians/People/Blois|Ralph Blois]] (at 593), as Jerome Buonaparte, King of Westphalia
#[[Social Victorians/People/Fitzgerald|Amelia, Lady FitzGerald]] (at 599), as Marie Joséphe, Queen of Poland, A.D. 1737
#Prince [[Social Victorians/People/Duleep Singh|Victor Duleep Singh]] (at 558), as Akbar
#[[Social Victorians/People/Kintore|Sydney, Countess Kintore]] (at 608), as Jane, Duchess of Gordon
#Lady [[Social Victorians/People/Kintore|Hilda Keith-Falconer]] (at 677), as Lady Susan Gordon
#Madame [[Social Victorians/People/Baudon de Mony|Baudon de Mony]] (at 568), as Princess of Navarre
#Monsieur [[Social Victorians/People/Baudon de Mony|Baudon de Mony]] (at 567), as a Louis-XIII Musketeer
#Mademoiselle [[Social Victorians/People/de Alcalo Galiano|de Alealo Galiano]] (at 631), as the Queen of the Fairies
#Mademoiselle [[Social Victorians/People/de Alcalo Galiano|Consuelo de Alealo Galiano]] (at 632), as Veure de Pierrot
#Blanche, [[Social Victorians/People/Coventry#Blanche, Countess of Coventry|Countess of Coventry]] (at 559), as an earlier Countess of Coventry
#Lady [[Social Victorians/People/Coventry#Lady Anne Coventry|Anne Coventry]] (at 560, as Serena
#Lady [[Social Victorians/People/Coventry#Lady Dorothy Coventry|Dorothy Coventry]] (at 561), also as Serena
#Rose Towneley-Bertie, [[Social Victorians/People/Norreys|Lady Norreys]] (at 680), as a Paysanne Galante from the time of Louis XVI
#[[Social Victorians/People/Lukach|Joseph Harry Lukach]] (at 685), as Henri de Rohan
#Alice Emily White Coke, Viscountess Coke (at 643), in 18th-century dress
#Hon. [[Social Victorians/People/Stanley#Hon. Ferdinand Charles Stanley|F. C. (Ferdinand Charles) Stanley]] (at 251), as a Grenadier Guard officer, 1660
#Lord [[Social Victorians/People/Stanley#Lord Stanley and Lady A. Stanley|Edward Stanley]] (at 187), as a Grenadier Guard officer, 1660
#[[Social Victorians/People/Jeune#Madeline Cecilia Carlyle Stanley|Madeline Cecilia Carlyle Stanley]] (at 552), accompanying [[Social Victorians/People/Jeune|Sir Francis Jeune]] and [[Social Victorians/People/Jeune|Lady Jeune]], as Lady Hopeton, after a miniature by Cosway
#Henry William Crichton, [[Social Victorians/People/Crichton#Lord Crichton|Viscount Crichton]] (at 646), in a costume of the Empire period
#[[Social Victorians/People/Ellis#Major-General Ellis|General Ellis]] (at 654), as an Elizabethan noble
#Lady [[Social Victorians/People/Durham#Lady Anne Lambton|Anne Lambton]] (at 659), as Mme. de Longueville, Louis XIII period
#Mrs. [[Social Victorians/People/Burton#Nellie Lisa Baillie and Colonel James Evan Bruce Baillie of Dochfour|Nellie Lisa Baillie]] (at 667), from the family group by Gainsborough
#Mrs. [[Social Victorians/People/Burton#Mr. and Mrs. Hamar Bass|Hamar (Louisa) Bass]] (at 439), from picture at Chesterfield House
#Miss [[Social Victorians/People/Burton#Jane Thornewill|Jane Thornewill]] (at 664), in a costume of the Georgian era
#Mrs. [[Social Victorians/People/Sneyd|Mary Evelyn Ellis Sneyd]] (at 667), as a Venetian
#Mr. [[Social Victorians/People/Foley|Foley]] (at 690), as a Hussar of the Napoleonic era
#Mr. [[Social Victorians/People/Crawley|E. Crawley]] (at 692), as a gentleman of the period of Charles I
#Mr. [[Social Victorians/People/Carter|J. Carter]] (at 697), as a Courtier of Elizabeth
#Lady [[Social Victorians/People/Wolseley|Louisa Wolseley]] (at 541), in an 18th-century dress (?)
#Edward Villiers, [[Social Victorians/People/Villiers#Edward Villiers, 5th Earl Clarendon|5th Earl Clarendon]] (at 65), as Villiers, Viscount Grandison, after portrait by Vandyke
#[[Social Victorians/People/Souza Correa|M. de Souza Correa]] (at 178), as a Knight Templar, XIV Century
#Margaret Child-Villiers, [[Social Victorians/People/Jersey#Lord and Lady Jersey|Countess of Jersey]] (at 432), as Anne of Austria
#Mary, Countess Minto (at 544) was Princess [[Social Victorians/People/Minto#Mary, Countess of Minto|Andrillon]]
#Lady [[Social Victorians/People/Pembroke#Lady Beatrix Herbert|Beatrix Herbert]] (at 648), as Signora Bacelli after Gainsborough
#Mrs. [[Social Victorians/People/Chamberlain#Mrs. Mary Chamberlain|Mary Chamberlain]] (at 491), as Madame d'Epinay
#Windham Thomas Wyndham-Quin, [[Social Victorians/People/Dunraven#Earl and Countess Dunraven|4th Earl of Dunraven and Mount-Earl]] (at 199), as Cardinal Mazarin
#Lady [[Social Victorians/People/Ancaster#Lady Evelyn Ewart|Evelyn Ewart]] (at 401), as the Duchess of Ancaster, Mistress of the Robes to Queen Charlotte, 1757, after a picture by Hudson
#Lady [[Social Victorians/People/Buchan#Lord and Lady Cardross|Rosalie Cardross]] (at 276), as La Duchesse de Lavis
#Lady [[Social Victorians/People/Duncombe|Florence Duncombe]] (at 456), as a Lady of the Court of Marie Stuart
#Lord [[Social Victorians/People/Gordon-Lennox#Lord Algernon Gordon Lennox|Algernon Gordon Lennox]] (at 623), as a Grenadier Guard officer
#Lady [[Social Victorians/People/Cole|Florence Cole]] (at 239), as Hortense Beauharnais
#Mr. [[Social Victorians/People/Myddleton-Biddulph|Algernon Myddleton Biddulph]] (at 268), as Count Soltykoff or Saltykov

== Notes and Questions ==

# Work this info in:

&lt;blockquote&gt;There is intense excitement (says a lady correspondent) about the Duchess of Devonshire's historical and fancy dress ball to take place to-night. One of the prettiest of Princesses, daughter of a lovely Irish mother, goes as Queen of Sheba, her sister representing an Ethiopian attendant. An illustrious personage is to head the list of old-world knights, and a beautiful Marchioness is to represent Guinevere, her fair young daughter going as Elaine. A most lovely lady is to personate Queen Marie Thérèse, surrounded by her Court. There is to be a procession of young girls dressed after Cosway's miniatures, and an Elizabethan quadrille is to be danced, in which the Virgin Queen herself is to appear, as well as Essex, Raleigh, Shakespeare, and other well-known characters. Another quadrille will be made up of ladies and gentlemen costumed after the style of Catherine II.'s Russian Court, but none will be more pictorially effective than that in which Catherine de Medici will appear, some of the gentlemen representing Henri II., Francis II., Charles IX., Henri III., Gaspard de Collini, Comte de la Marck, and the Duc de Guise.&lt;ref name=&quot;:4&quot;&gt;&quot;This Morning's News.&quot; ''London Daily News'' 2 July 1897, Friday: 5 [of 10], Col. 3B. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970702/026/0005.&lt;/ref&gt;&lt;/blockquote&gt;

== Bibliography for Courts ==
* Drew-Smythe David. &quot;The Duchess of Devonshire's Ball, 1897.&quot; [&quot;principal names in the courts&quot;] http://www.zipworld.com.au/~lnbdds/home/rah/dodbcourts.htm (accessed July 2017). Based on the report in the ''Times''.
* Ross, Sarah. &quot;The Devonshire House Ball (1897): A Guest-list of Society.&quot; ''Pax Victoriana: The Age of Victoria: The Long 19th Century in Literature and Everyday Life'' http://paxvictoriana.tumblr.com/post/101421946818/the-devonshire-house-ball-1897-a-guest-list-of (accessed July 2017). Based on the report in the ''Times''.
== Footnotes ==
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  <page>
    <title>Complex cube root</title>
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      <timestamp>2026-06-11T09:12:53Z</timestamp>
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        <username>ThaniosAkro</username>
        <id>2805358</id>
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      <text bytes="32194" sha1="anjwe4t7wqn5nrqi7b8aav0nhf3ndyk" xml:space="preserve">=Introduction=
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[[File:0419polarDiagram.png|thumb|400px|'''Complex number W = complex number w³.'''
&lt;/br&gt;
Origin at point &lt;math&gt;(0,0)&lt;/math&gt;.&lt;/br&gt;
&lt;math&gt;w_{real}, W_{real}&lt;/math&gt; parallel to &lt;math&gt;X&lt;/math&gt; axis.&lt;/br&gt;
&lt;math&gt;w_{imag}, W_{imag}&lt;/math&gt; parallel to &lt;math&gt;Y&lt;/math&gt; axis.&lt;/br&gt;
&lt;math&gt;w_{real} = 1.2;\ w_{imag} = 0.5;\ w_{mod} = \sqrt{1.2^2 + 0.5^2} = 1.3.&lt;/math&gt;&lt;/br&gt;
&lt;math&gt;W_{real} = 0.828;\ ;\ W_{imag} = 2.035;&lt;/math&gt;&lt;/br&gt;
&lt;math&gt;W_{mod} = \sqrt{0.828^2 + 2.035^2} = 2.197.&lt;/math&gt;&lt;/br&gt;
&lt;math&gt;W_{mod} = w_{mod}^3 = 1.3^3 = 2.197.&lt;/math&gt;&lt;/br&gt;
&lt;math&gt;W_{\phi} = 3 w_{\phi}.&lt;/math&gt;
By cosine triple angle formula:&lt;/br&gt;
&lt;math&gt;\cos W_{phi} = 4\cdot(\frac{1.2}{1.3})^3 - 3\cdot \frac{1.2}{1.3} = \frac{828}{2197} = \frac{W_{real}}{W_{mod}}.&lt;/math&gt;
&lt;/br&gt;
See &quot;3 cube roots of W&quot; in [[Complex_cube_root#Gallery | Gallery]] below.
]]
Let complex numbers &lt;math&gt;W =&lt;/math&gt; a &lt;math&gt;+&lt;/math&gt; b&lt;math&gt;\cdot i&lt;/math&gt; and
&lt;math&gt;w =&lt;/math&gt; p &lt;math&gt;+&lt;/math&gt; q&lt;math&gt;\cdot i.&lt;/math&gt;
Let &lt;math&gt;W = w^3.&lt;/math&gt;

When given &lt;math&gt;a, b,&lt;/math&gt; aim of this page is to calculate &lt;math&gt;p, q.&lt;/math&gt;


In the diagram complex number &lt;math&gt;w = p + qi = w_{real} + i\cdot w_{imag} = w_{mod}(\cos w_{\phi} + i\cdot \sin w_{\phi}),&lt;/math&gt; where

* &lt;math&gt;w_{real}, w_{imag}&lt;/math&gt; are the real and imaginary components of &lt;math&gt;w,&lt;/math&gt; the rectangular components.

* &lt;math&gt;w_{mod}, w_{\phi}&lt;/math&gt; are the modulus and phase of &lt;math&gt;w,&lt;/math&gt; the polar components.

Similarly, &lt;math&gt;W_{real}, W_{imag}, W_{mod}, W_{\phi}&lt;/math&gt; are the corresponding components of &lt;math&gt;W.&lt;/math&gt;


&lt;math&gt;W = w^3 = ( w_{mod}(\cos w_{\phi} + i\cdot \sin w_{\phi}) )^3&lt;/math&gt;

&lt;math&gt;= w_{mod}^3(\cos (3 w_{\phi}) + i\cdot \sin (3 w_{\phi}))&lt;/math&gt;

&lt;math&gt;= W_{mod}(\cos (W_{\phi}) + i\cdot \sin (W_{\phi}))&lt;/math&gt;

&lt;math&gt;= W&lt;/math&gt;


There are 2 significant calculations:

&lt;math&gt;w_{mod} = \sqrt[3]{W_{mod}}&lt;/math&gt; and

&lt;math&gt;\cos w_{\phi} = \cos \frac{W_{\phi}}{3}.&lt;/math&gt;
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=Implementation=
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==Cos φ from cos 3φ==
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[[File:1008Cos3A.png|thumb|400px|'''
Graph of &lt;math&gt;\cos(3A) = 4 \cos^3(A) - 3 \cos(A)&lt;/math&gt;'''
&lt;/br&gt;
Formula and/or graph are used to calculate &lt;math&gt;\cos(A)&lt;/math&gt; if &lt;math&gt;\cos(3A)&lt;/math&gt; is given.
]]
The cosine triple angle formula is:
&lt;math&gt;\cos (3\theta) = 4 \cos^3\theta - 3 \cos\theta.&lt;/math&gt; This formula, of form &lt;math&gt;y = 4 x^3 - 3 x,&lt;/math&gt; permits &lt;math&gt;\cos (3\theta)&lt;/math&gt; to be calculated if &lt;math&gt;\cos\theta&lt;/math&gt; is known.

If &lt;math&gt;\cos (3\theta)&lt;/math&gt; is known and the value of &lt;math&gt;\cos\theta&lt;/math&gt; is desired, this identity becomes:
&lt;math&gt;4 \cos^3\theta - 3 \cos\theta - \cos (3\theta) = 0.&lt;/math&gt; &lt;math&gt;\cos\theta&lt;/math&gt; is the solution of this cubic equation.

In fact this equation has three solutions, the other two being &lt;math&gt;\cos (\theta \pm 120^\circ).&lt;/math&gt;

&lt;math&gt;\cos (3(\theta \pm 120^\circ)) = \cos (3\theta \pm 360^\circ) = \cos (3\theta).&lt;/math&gt;


It is sufficient to calculate only &lt;math&gt;\cos\theta&lt;/math&gt; from &lt;math&gt;\cos 3\theta,&lt;/math&gt; accomplished by the following code:
&lt;syntaxhighlight lang=python&gt;
# python code

cosAfrom_cos3Adebug = 0

def cosAfrom_cos3A(cos3A) :
    cos3A = Decimal(str(cos3A))+0
    if 1 &gt;= cos3A &gt;= -1 : pass
    else :
        print ('cosAfrom_cos3A(cos3A) : cos3A not in valid range.')
        return None
    '''
    if cos3A == 0 :
        A = 90 and cos3A = cosA
    if cos3A == 1 :
        A = 0 and cos3A = cosA
    if cos3A == -1 :
        A = 180 and cos3A = cosA
'''
    if cos3A in (0,1,-1) : return cos3A

    # From the cosine triple angle formula:
    a,b,c,d = 4,0,-3,-cos3A

    # prepare for Newton's method.
    if d &lt; 0 : x = Decimal(1)
    else : x = -Decimal(1)
    count = 31; L1 = [x]; almostZero = Decimal('1e-' + str(prec-2))

    # Newton's method:
    while 1 :
        count -= 1
        if count &lt;= 0 :
            print ('cosAfrom_cos3A(cos3A): count expired.')
            break
        y = a*x*x*x + c*x + d
        if cosAfrom_cos3Adebug :
            print ('cosAfrom_cos3A(cos3A) : x,y =',x,y)
        if abs(y) &lt; almostZero : break
        slope = 12*x*x + c
        delta_x = y/slope
        x -= delta_x
        if x in L1[-1:-5:-1] :
            # This value of x has been used previously.
            print ('cosAfrom_cos3A(cos3A): endless loop detected.')
            break
        L1 += [x]

    return x
&lt;/syntaxhighlight&gt;
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When &lt;math&gt;x == 1,&lt;/math&gt; slope &lt;math&gt;= 12x^2 - 3 = 9.&lt;/math&gt; Within area of interest, maximum absolute value of slope &lt;math&gt;= 9,&lt;/math&gt;
a rather small value for slope.

Consequently, with only 9 passes through loop, Newton's method produces a result accurate to 200 places of decimals .
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There are 3 conditions, any 1 of which terminates the loop:

* &lt;code&gt;abs(y)&lt;/code&gt; very close to 0 (normal termination).

* count expired.

* endless loop detected with the same value of &lt;code&gt;x&lt;/code&gt; repeated.
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==Calculation of cube roots of W==
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===Calculation of 1 root===

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

def complexCubeRoot (a,b) :
    '''
p,q = complexCubeRoot (a,b) where:
* a,b are rectangular coordinates of complex number a + bi.
* (p + qi)^3 = a + bi.
'''

    a = Decimal(str(a))
    b = Decimal(str(b))

    if a == b == 0 : return (Decimal(0), Decimal(0))
    if a == 0 : return (Decimal(0), -simpleCubeRoot(b))
    if b == 0 : return (simpleCubeRoot(a), Decimal(0))

    Wmod = (a*a + b*b).sqrt()
    wmod = simpleCubeRoot (Wmod)
    cosWφ = a/Wmod
    coswφ = cosAfrom_cos3A(cosWφ)
    p = coswφ * wmod ; P = p**2 # p = wreal
    Q = wmod*wmod - P # Q = q**2
    # 3ppq - Qq = b
    # (3pp - Q)q = b
    q = b/(3*P - Q) # wimag

    return p,q
&lt;/syntaxhighlight&gt;
For function &lt;code&gt;simpleCubeRoot()&lt;/code&gt; see [[ Cube_root#Implementation | Cube_root#Implementation ]]
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===Calculation of 3 roots===

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See [[ Python_Concepts/Numbers#Cube_roots_of_1_simplified | Cube roots of unity. ]]

The cube roots of unity are :
&lt;math&gt;1, \frac{-1 \pm i\sqrt{3}}{2}.&lt;/math&gt;

When &lt;math&gt;r_0 = \sqrt[3]{W}&lt;/math&gt; is known, the other 2 cube roots are:

* &lt;math&gt;r_1 = r_0 \cdot \frac{-1 + i\sqrt{3}}{2}&lt;/math&gt;

* &lt;math&gt;r_2 = r_0 \cdot \frac{-1 - i\sqrt{3}}{2}&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# python code
def complexCubeRoots (a,b) :
    '''
r0,r1,r2 = complexCubeRoots (a,b) where:
* a,b are rectangular coordinates of complex number a + bi.
* (p + qi)^3 = a + bi.
* r0 = (p0,q0)
* r1 = (p1,q1)
* r2 = (p2,q2)
'''
    p,q = complexCubeRoot (a,b)
    r3 = Decimal(3).sqrt() # Square root of 3.
    pr3,qr3 = p*r3, q*r3
#  r1 = ((-p-q*r)/2, (p*r - q)/2)
#  r2 = ((-p+q*r)/2, (-p*r - q)/2)
    r0 = (p,q)
    r1 = ((-p-qr3)/2, (pr3 - q)/2)
    r2 = ((-p+qr3)/2, (-pr3 - q)/2)
    return r0,r1,r2
&lt;/syntaxhighlight&gt;
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=Testing results=

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&lt;math&gt;(p + q\cdot i)^3 = p^3 - 3pq^2 + (3p^2q - q^3)\cdot i = a + b\cdot i.&lt;/math&gt;

&lt;math&gt;a = p^3 - 3pq^2;\ b = 3p^2q - q^3.&lt;/math&gt;
&lt;syntaxhighlight lang=python&gt;
# Python code used to test results of code above,

def complexCubeRootsTest (a,b) :
    print ('\n++++++++++++++++++++')
    print ('a,b =', a,b)
    almostZero = Decimal('1e-' + str(prec-5))
    r0,r1,r2 = complexCubeRoots (a,b)
    for root in (r0,r1,r2) :
        p,q = root
        print ('  pq =',(p), (q))
        a_,b_ = (p*p*p - 3*p*q*q), (3*p*p*q - q*q*q)
        if a :
            v1 = abs((a_-a)/a)
            if v1 &gt; almostZero : print ('error *',a_,a,v1)
        else :
            v1 = abs(a_)
            if v1 &gt; almostZero : print ('error !',a_,a,v1)
        if b :
            v1 = abs((b_-b)/b)
            if v1 &gt; almostZero : print ('error &amp;',b_,b,v1)
        else :
            v1 = abs(b_)
            if v1 &gt; almostZero : print ('error %',b_,b,v1)
    return r0,r1,r2
    
import decimal
Decimal = D = decimal.Decimal
prec = decimal.getcontext().prec # precision

cosAfrom_cos3Adebug = 1

for p in range (-10,11,1) :
    for q in range (-10,11,1) :
        a = p*p*p - 3*p*q*q
        b = 3*p*p*q - q*q*q
        complexCubeRootsTest(a,b)
&lt;/syntaxhighlight&gt;
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=Gallery=
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&lt;gallery&gt;
File:0421cubeRootsOf8i.png|&lt;small&gt;Cube roots of 8i.&lt;/small&gt;
File:0422cubeRootsOfm8.png|&lt;small&gt;Cube roots of -8.&lt;/small&gt;
File:0424_3cubeRoots01.png|&lt;small&gt;3 cube roots of W.&lt;/small&gt;
&lt;/gallery&gt;
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=Method #2 (Graphical)=
==Introduction==
[[File:0406_2curves01a.png|thumb|400px|'''Graphs of 2 curves showing complex cube roots as points of intersection of the 2 curves.''']]
Let complex number &lt;math&gt;w = p + qi.&lt;/math&gt;

Then &lt;math&gt;W = w^3 = (p + qi)^3 = p^3 - 3pq^2 + (3p^2q - q^3)i.&lt;/math&gt;

Let &lt;math&gt;W = a + bi.&lt;/math&gt;

Then:

&lt;math&gt;a = p^3 - 3pq^2&lt;/math&gt; and

&lt;math&gt;b = 3p^2q - q^3.&lt;/math&gt;


When &lt;math&gt;W&lt;/math&gt; is given and &lt;math&gt;w&lt;/math&gt; is desired,
&lt;math&gt;w&lt;/math&gt; may be calculated from the solutions of 2 simultaneous equations:

&lt;math&gt;p^3 - 3pq^2 - a = 0\ \dots\ (1)&lt;/math&gt; and

&lt;math&gt;3p^2q - q^3 - b = 0\ \dots\ (2).&lt;/math&gt;


For example, let &lt;math&gt;W = (39582 + 3799i).&lt;/math&gt;


Then equations &lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2)&lt;/math&gt; become (for graphical purposes):

&lt;math&gt;x^3 - 3xy^2 - 39582 = 0\ \dots\ (3),&lt;/math&gt; black curve in diagram, and

&lt;math&gt;3x^2y - y^3 - 3799 = 0\ \dots\ (4),&lt;/math&gt; red curve in diagram.


Three points of intersection of red and black curves are:

&lt;math&gt;(-18, 29),&lt;/math&gt;

&lt;math&gt;(34.11473670974872, 1.0884572681198943),&lt;/math&gt; and

&lt;math&gt;(-16.11473670974872, -30.088457268119896),&lt;/math&gt;

interpreted as the three complex cube roots of &lt;math&gt;W,&lt;/math&gt; namely:

&lt;math&gt;w_0 = (-18 + 29i),&lt;/math&gt;

&lt;math&gt;w_1 = (34.11473670974872 + 1.0884572681198943i)&lt;/math&gt; and

&lt;math&gt;w_2 = (-16.11473670974872 - 30.088457268119896i).&lt;/math&gt;
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Proof:
&lt;syntaxhighlight lang=python&gt;
# python code
# Each cube root cubed.
&gt;&gt;&gt; w0 = (-18 + 29j)
&gt;&gt;&gt; w1 = (34.11473670974872 + 1.0884572681198943j)
&gt;&gt;&gt; w2 = (-16.11473670974872 - 30.088457268119896j)
&gt;&gt;&gt; for w in (w0,w1,w2) : w**3
... 
(39582+3799j)
(39582+3799j)
(39582+3799j)
# The moduli of all 3 cube roots.
&gt;&gt;&gt; for w in (w0,w1,w2) : (w.real**2 + w.imag**2) ** 0.5
... 
34.132096331752024
34.132096331752024
34.132096331752024
&lt;/syntaxhighlight&gt;
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==Preparation==

This method depends upon selection of most appropriate quadrant.

In the example above, 
quadrant &lt;math&gt;2&lt;/math&gt; is chosen because any non-zero positive value of &lt;math&gt;y&lt;/math&gt; intersects red curve and
any non-zero negative value of &lt;math&gt;x&lt;/math&gt; intersects black curve.

Figures 1-4 below show all possibilities of &lt;math&gt;\pm a&lt;/math&gt; and &lt;math&gt;\pm b.&lt;/math&gt;
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&lt;gallery&gt;
File:0406_2curves01a.png|&lt;small&gt;Figure 1. When both &lt;math&gt;a, b&lt;/math&gt; are positive, quadrant 2 is chosen.&lt;/br&gt;&lt;math&gt;(-18+29i)^3&lt;/math&gt;&lt;math&gt; = (39582+3799i)&lt;/math&gt;&lt;/small&gt;
File:0411np01.png|&lt;small&gt;Figure 2. When &lt;math&gt;a&lt;/math&gt; is positive and &lt;math&gt;b&lt;/math&gt; is negative, quadrant 3 is chosen.&lt;/br&gt;&lt;math&gt;(-18-29i)^3&lt;/math&gt;&lt;math&gt; = (39582-3799i)&lt;/math&gt;&lt;/small&gt;
File:0411pn01.png|&lt;small&gt;Figure 3. When &lt;math&gt;a&lt;/math&gt; is negative and &lt;math&gt;b&lt;/math&gt; is positive, quadrant 1 is chosen.&lt;/br&gt;&lt;math&gt;(18+29i)^3&lt;/math&gt;&lt;math&gt; = (-39582+3799i)&lt;/math&gt;&lt;/small&gt;
File:0406_2curves00a.png|&lt;small&gt;Figure 4. When both &lt;math&gt;a, b&lt;/math&gt; are negative, quadrant 4 is chosen.&lt;/br&gt;&lt;math&gt;(18-29i)^3&lt;/math&gt;&lt;math&gt; = (-39582-3799i)&lt;/math&gt;&lt;/small&gt;
&lt;/gallery&gt;
Lines &lt;math&gt;y = x&lt;/math&gt; and &lt;math&gt;y = -x&lt;/math&gt; (not shown) intersect:

* red curves at points where slope of red curve is vertical, and

* black curves at points where slope of black curve is horizontal.
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==Description of method==
===Four points===
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[[File:0406_4points.png|thumb|400px|'''Graphs of 2 curves showing 4 points that enclose one of the complex cube roots.'''&lt;/br&gt;
Area enclosed by the 4 points becomes progressively smaller and smaller until the point of intersection is identified.
]]
Assume that &lt;math&gt;W = -39582 - 3799i,&lt;/math&gt; in which case both &lt;math&gt;a, b&lt;/math&gt; are negative and quadrant &lt;math&gt;4&lt;/math&gt; is chosen.

In quadrant &lt;math&gt;4&lt;/math&gt; any non-zero positive value of x intersects black curve and any non-zero negative value of y intersects red curve.


Choose any convenient negative, non-zero value of &lt;math&gt;y.&lt;/math&gt;

Let &lt;math&gt;y = -18.&lt;/math&gt;

Using this value of &lt;math&gt;y,&lt;/math&gt; calculate coordinates of point &lt;math&gt;A&lt;/math&gt; on red curve.

Using &lt;math&gt;x&lt;/math&gt; coordinate of point &lt;math&gt;A,&lt;/math&gt; calculate coordinates of point &lt;math&gt;B&lt;/math&gt;
on black curve.

Using &lt;math&gt;y&lt;/math&gt; coordinate of point &lt;math&gt;B,&lt;/math&gt; calculate coordinates of point &lt;math&gt;C&lt;/math&gt;
on red curve.

Using &lt;math&gt;x&lt;/math&gt; coordinate of point &lt;math&gt;C,&lt;/math&gt; calculate coordinates of point &lt;math&gt;D&lt;/math&gt;
on black curve.


Points &lt;math&gt;A, B, C, D&lt;/math&gt; enclose the point of intersection of the 2 curves.
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===Point of intersection===
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[[File:0406intersection.png|thumb|400px|'''Graphs of 2 curves showing complex cube root enclosed by four points &lt;math&gt;A, B, C, D&lt;/math&gt;.'''&lt;/br&gt;
Point &lt;math&gt;E,&lt;/math&gt; intersection of lines &lt;math&gt;AC, BD&lt;/math&gt; is close to complex cube root, and is starting point for next iteration.
]]
Calculate equations of lines &lt;math&gt;AC, BD.&lt;/math&gt;

Calculate coordinates of point &lt;math&gt;E,&lt;/math&gt; intersection of lines &lt;math&gt;AC, BD.&lt;/math&gt;


Point &lt;math&gt;E&lt;/math&gt; is used as starting point for next iteration.


Area of quadrilateral &lt;math&gt;ABCD&lt;/math&gt; becomes smaller and smaller until complex cube root, intersection of red and black curves, is identified.
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==Implementation==
===Initialization===
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&lt;syntaxhighlight lang=python&gt;
# python code

import decimal
D = decimal.Decimal

precision = decimal.getcontext().prec = 100

useDecimal = 1

Tolerance = 1e-14
if useDecimal :
    Tolerance = D('1e-' + str(decimal.getcontext().prec - 2))


def line_mc (point1,point2) :
    '''
m,c = line_mc (point1,point2)
where y = mx + c.
'''
    x1,y1 = point1
    x2,y2 = point2
    m = (y2-y1) / (x2-x1)
    # y = mx + c
    c = y1 - m*x1
    return m,c


def intersectionOf2Lines (line1, line2) :
    m1,c1 = line1
    m2,c2 = line2
    # y = m1x + c1
    # y = m2x + c2
    # m1x + c1 = m2x + c2
    # m1x - m2x = c2 - c1
    x = (c2-c1)/(m1-m2)
    y = m1*x + c1
    return x,y

def almostEqual (v1,v2,tolerance = Tolerance) :
    '''
status = almostEqual (v1,v2)

For floats, tolerance is 1e-14.

1234567.8901234567 and
1234567.8901234893
are not almostEqual.

1234567.8901234567 and
1234567.8901234593
are almostEqual.
'''
    return abs(v1-v2) &lt; tolerance*abs((v1+v2)/2)
&lt;/syntaxhighlight&gt;
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===Function two_points()===
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[[File:0406_4points.png|thumb|400px|'''Graphs of 2 curves showing 4 points that enclose one of the complex cube roots.'''&lt;/br&gt;
On first invocation function two_points() returns points &lt;math&gt;A, B.&lt;/math&gt;&lt;/br&gt;
On second invocation function two_points() returns points &lt;math&gt;C, D.&lt;/math&gt;&lt;/br&gt;
If distance &lt;math&gt;AB&lt;/math&gt; or distance &lt;math&gt;CD&lt;/math&gt; is very small, function two_points() returns status True.
]]
&lt;syntaxhighlight lang=python&gt;
def two_points (y,a,b,quadrant) :
    '''
[point1,point2],status = two_points (y,a,b)
'''

    L1 = [] ; yInput = y
    if 0 :
        print ('two_points():')
        s1 = '  a,b,y' ; print (s1, eval(s1))

# a = ppp - pqq - 2pqq = ppp - 3pqq (1)
# b = 2ppq + ppq - qqq = 3ppq - qqq (2)
#
# Using (2)
# 3xxy - yyy = b
#
#     yyy + b
# X = ----------
#        3y
    X = (y**3 + b) / (3*y)

    if isinstance(X,D) : x = X.sqrt()
    else : x = X ** 0.5
    if quadrant in (2,3) : x *= -1
    L1 += [(x,y)]

# Using (1)
# xxx - 3xyy = a
#
#     xxx - a
# Y = -----------
#        3x
#
    Y = (x**3 - a) / (3*x)

    if isinstance(Y,D) : y = Y.sqrt()
    else : y = Y ** 0.5
    if quadrant in (3,4) : y *= -1

    L1 += [(x,y)]

    return L1, almostEqual(y, yInput)
&lt;/syntaxhighlight&gt;
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===Function pointOfIntersection()===
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[[File:0406intersection.png|thumb|400px|'''Graphs of 2 curves showing complex cube root enclosed by four points &lt;math&gt;A, B, C, D&lt;/math&gt;.'''&lt;/br&gt;
From points &lt;math&gt;A, B, C, D&lt;/math&gt; function pointOfIntersection() calculates coordinates of point &lt;math&gt;E.&lt;/math&gt;&lt;/br&gt;
If distance &lt;math&gt;AB&lt;/math&gt; is very small, point &lt;math&gt;A&lt;/math&gt; is returned as equivalent to intersection of red and black curves.&lt;/br&gt;
If distance &lt;math&gt;CD&lt;/math&gt; is very small, point &lt;math&gt;C&lt;/math&gt; is returned as equivalent to intersection of red and black curves.
]]
&lt;syntaxhighlight lang=python&gt;
def pointOfIntersection(y,a,b,quadrant) :
    '''
pointE, status = pointOfIntersection(y,a,b)
y is Y coordinate of point A.
'''
#    print('\npointOfIntersection()')
    t1,status = two_points (y,a,b,quadrant)
    ptA,ptB = t1
    if status :
        # Distance between ptA and ptB is very small.
        # ptA is considered equivalent to the complex cube root.
        return ptA,status

    t2,status = two_points (ptB[1],a,b,quadrant)
    ptC,ptD = t2
    if status :
        # Distance between ptC and ptD is very small.
        # ptC is considered equivalent to the complex cube root.
        return ptC,status

    lineAC = line_mc (ptA,ptC)
    lineBD = line_mc (ptB,ptD)
    pointE = intersectionOf2Lines (lineAC, lineBD)
    return pointE,False
&lt;/syntaxhighlight&gt;
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===Execution===
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&lt;syntaxhighlight lang=python&gt;
def CheckMake_pq(a,b,x,y) :
    '''
p,q = CheckMake_pq(a,b,x,y)
p = x and q = y.
p,q are checked as valid within tolerance, and are reformatted slightly to improve appearance.
'''
#    print ('\nCheckMake_pq(a,b,x,y)')
#    s1 = '(a,b)' ; print (s1,eval(s1))
#    s1 = '   x'  ; print (s1,eval(s1))
#    s1 = '   y'  ; print (s1,eval(s1))
    P = x**2 ; Q = y**2 ; p=q=-1
    for p in (x,) :
        # a = ppp - 3pqq; a + 3pqq should equal ppp.
        if not almostEqual (P*p, a + 3*p*Q, Tolerance*100) : continue
        for q in  (y,) :
            # b = 3ppq - qqq; b + qqq should equal 3ppq
            if not almostEqual (3*P*q, b + q*Q) : continue
            # Following 2 lines improve appearance of p,q
            if useDecimal :
                # 293.00000000000000000000000000000000000000034 becomes 293
                p,q = [ decimal.Context(prec=precision-3).create_decimal(s).normalize()
                        for s in (p,q) ]
            else :
                # 123.99999999999923 becomes 124.0
                p,q = [ float(decimal.Context(prec=14).create_decimal(s)) for s in (p,q) ]
            return p,q
    # If code gets to here there is internal error.
    s1 = '   p'  ; print (s1,eval(s1))
    s1 = '   q'  ; print (s1,eval(s1))
    s1 = 'P*p, a + 3*p*Q' ; print (s1,eval(s1))
    s1 = '3*P*q, b + q*Q' ; print (s1,eval(s1))
    1/0


def ComplexCubeRoot (a,b, y = 100, count_ = 20) :
    '''
p,q = ComplexCubeRoot (a,b, y, count_)
(p+qi)**3 = (a+bi)
'''
    print ('\nComplexCubeRoot(): a,b,y,count_ =',a,b,y,count_)

    if useDecimal : y,a,b = [ D(str(v)) for v in (y,a,b) ]

    if a == 0 :
        if b == 0 : return 0,0
        if useDecimal : cubeRoot = abs(b) ** (D(1)/3)
        else : cubeRoot = abs(b) ** (1/3)
        if b &gt; 0 : return 0,-cubeRoot
        return 0,cubeRoot
    if b == 0 :
        if useDecimal : cubeRoot = abs(a) ** (D(1)/3)
        else : cubeRoot = abs(a) ** (1/3)
        if a &gt; 0 : return cubeRoot,0
        return -cubeRoot,0

    # Select most appropriate quadrant.
    if a &gt; 0: setx = {2,3}
    else: setx = {1,4}

    if b &gt; 0: sety = {1,2}
    else: sety = {3,4}

    quadrant, = setx &amp; sety
    # Make sign of y appropriate for this quadrant.
    if quadrant in (1,2) : y = abs(y)
    else : y = -abs(y)

    s1 = '    quadrant' ;    print (s1, eval(s1))

    for count in range (0,count_) :
        pointE,status = pointOfIntersection(y,a,b,quadrant)
        s1 = 'count,status' ; print (s1, eval(s1))
        s1 = '    pointE[0]' ;    print (s1, eval(s1))
        s1 = '    pointE[1]' ;    print (s1, eval(s1))
        x,y = pointE
        if status : break

    p,q = CheckMake_pq(a,b,x,y)

    return p,q
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}

==An Example==
{{RoundBoxTop|theme=2}}
&lt;syntaxhighlight lang=python&gt;
p,q = 18,-29
w0 = p + q*1j
W = w0**3
a,b = W.real, W.imag
s1 = '\na,b' ; print (s1, eval(s1))
print ('Calculate one cube root of W =', W)

p,q = ComplexCubeRoot (a, b, -18)
s1 = '\np,q' ; print (s1, eval(s1))

sign = ' + '
if q &lt; 0 : sign = ' - '
print ('w0 = ', str(p) ,sign, str(abs(q)),'i',sep='')
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
a,b (-39582.0, -3799.0)
Calculate one cube root of W = (-39582-3799j)

ComplexCubeRoot(): a,b,y,count_ = -39582.0 -3799.0 -18 20
    quadrant 4
count,status (0, False)
    pointE[0] 18.29530595866769796981147594954794157453427770441979517949705190002312860122683512802090262517713985
    pointE[1] -29.17605851188829785804056660826030733025475591125914094664311767722817017040959484906193571713185723
count,status (1, False)
    pointE[0] 18.00005338608833140244623119091867294731079673210031643698475740639013316464725974710029414039192178
    pointE[1] -29.00004871113589733281025047965490410760310240487028781197782902118493613318468122514940700337153822
count,status (2, False)
    pointE[0] 18.00000000000411724901243622639913339568271402799883998577879934397861645683113192688877413853607310
    pointE[1] -29.00000000000374389488957142528693701977412078931714190614174861872117809632376941851192785888688849
count,status (3, False)
    pointE[0] 18.00000000000000000000000002432197332441306371168312765664226520285586754485200564671348858119116700
    pointE[1] -29.00000000000000000000000002211642401405406173668734741733917293863102998696594080783163691859719835
count,status (4, False)
    pointE[0] 18.00000000000000000000000000000000000000000000000000000084875301166228708732099292145747224660296019
    pointE[1] -29.00000000000000000000000000000000000000000000000000000077178694502906372553368739653233322951192943
count,status (5, False)
    pointE[0] 18.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
    pointE[1] -29.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
count,status (6, True)
    pointE[0] 18
    pointE[1] -29.00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

p,q (Decimal('18'), Decimal('-29'))
w0 = 18 - 29i
&lt;/syntaxhighlight&gt;
{{RoundBoxBottom}}
=Method #3 (Algebraic)=

==Introduction==
Let complex number &lt;math&gt;w = p + qi.&lt;/math&gt;

Then &lt;math&gt;W = w^3 = (p + qi)^3 = p^3 - 3pq^2 + (3p^2q - q^3)i.&lt;/math&gt;

Let &lt;math&gt;W = a + bi.&lt;/math&gt;

Then:

&lt;math&gt;a = p^3 - 3pq^2\ \dots\ (1)&lt;/math&gt; and

&lt;math&gt;b = 3p^2q - q^3\ \dots\ (2).&lt;/math&gt;


When &lt;math&gt;W&lt;/math&gt; is given and &lt;math&gt;w&lt;/math&gt; is desired,
&lt;math&gt;w&lt;/math&gt; may be calculated from the solutions of 2 simultaneous equations
&lt;math&gt;(1)&lt;/math&gt; and &lt;math&gt;(2),&lt;/math&gt; where &lt;math&gt;a,b&lt;/math&gt; are
known values, and &lt;math&gt;p,q&lt;/math&gt; are desired.

==Implementation==

&lt;math&gt;(1)&lt;/math&gt; squared: &lt;math&gt;a^2 = p^6 - 6p^4q^2\ + 9p^2q^4&lt;/math&gt;

&lt;math&gt;p^2 p^2 p^2 - (6) p^2 p^2 q^2\ + (9)p^2q^4 - a^2\dots\ (1a)&lt;/math&gt;


&lt;math&gt;(1a) * (3q)^3:&lt;/math&gt;

&lt;math&gt;3qp^2\ 3qp^2\ 3qp^2 - (6) 3q\ 3qp^2\ 3qp^2\ q^2\ + (9)3q3q\ 3qp^2\ q^4 - 3q3q3qa^2\dots\ (1b)&lt;/math&gt;

From &lt;math&gt;(2):\ 3qp^2 = b + q^3\ \dots\ (2a)&lt;/math&gt;


For &lt;math&gt;(3qp^2)&lt;/math&gt; in &lt;math&gt;(1b)&lt;/math&gt; substitute &lt;math&gt; (b + q^3) : &lt;/math&gt;

&lt;math&gt;(b + q^3) (b + q^3) (b + q^3) - (6) 3q (b + q^3) (b + q^3) q^2\ + (9)3q3q (b + q^3) q^4 - 3q3q3qa^2\dots\ (1c)&lt;/math&gt;


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

&lt;math&gt;f(Q) = sQ^3 + tQ^2 + uQ + v\ \dots\ (3)&lt;/math&gt; where:

&lt;math&gt; Q  =  q ^ 3 &lt;/math&gt;

&lt;math&gt;s = 64&lt;/math&gt;

&lt;math&gt;t = 48b&lt;/math&gt;

&lt;math&gt;u = -(15b^2 + 27a^2)&lt;/math&gt;

&lt;math&gt;v = b^3&lt;/math&gt;


Calculate one real root of &lt;math&gt;(3):\ Q_1&lt;/math&gt;

&lt;math&gt;q_1 = \sqrt [3] {Q_1} &lt;/math&gt;

From &lt;math&gt;(2a):\ p_1 = \sqrt{\frac{b + Q_1}{3q_1}}&lt;/math&gt;


Check &lt;math&gt;p_1&lt;/math&gt; against &lt;math&gt;(1)&lt;/math&gt; to resolve ambiguity of sign of &lt;math&gt;p_1.&lt;/math&gt;


&lt;math&gt;p_1 + q_1 i&lt;/math&gt; is one cube root of &lt;math&gt;W.&lt;/math&gt;

{{RoundBoxTop|theme=2}}
&lt;math&gt;p&lt;/math&gt; may be calculated without ambiguity as follows:


&lt;math&gt;a = p^3 - 3pq^2\ \dots\ (1)&lt;/math&gt; and

&lt;math&gt;b = 3p^2q - q^3\ \dots\ (2).&lt;/math&gt;

From &lt;math&gt;(1):\ p^3 - 3q^2p - a = 0\ \dots\ (3)&lt;/math&gt;

From &lt;math&gt;(2):\ 3qp^2 - (Q + b) = 0\ \dots\ (4)&lt;/math&gt;


Let:

&lt;math&gt;A = -3q^2&lt;/math&gt;

&lt;math&gt;B = -a&lt;/math&gt;

&lt;math&gt;C = 3q&lt;/math&gt;

&lt;math&gt;D = -(Q + b)&lt;/math&gt;

{| class=&quot;wikitable&quot;
|-
| From &lt;math&gt;(3):&lt;/math&gt; || &lt;math&gt;p^3&lt;/math&gt;|| || &lt;math&gt;+Ap&lt;/math&gt;  || &lt;math&gt;+B&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (5)&lt;/math&gt;
|-
| From &lt;math&gt;(4):&lt;/math&gt; || || &lt;math&gt;Cp^2&lt;/math&gt;|| &lt;math&gt;&lt;/math&gt;  || &lt;math&gt;+D&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (6)&lt;/math&gt;
|-
| &lt;math&gt;(5)*D&lt;/math&gt; || &lt;math&gt;Dp^3&lt;/math&gt;|| || &lt;math&gt;+DAp&lt;/math&gt;  || &lt;math&gt;+DB&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (7)&lt;/math&gt;
|-
| &lt;math&gt;(6)*B&lt;/math&gt; || || &lt;math&gt;BCp^2&lt;/math&gt;|| &lt;math&gt;&lt;/math&gt;  || &lt;math&gt;+BD&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (8)&lt;/math&gt;
|-
| &lt;math&gt;(7)-(8)&lt;/math&gt; || &lt;math&gt;Dp^3&lt;/math&gt;||&lt;math&gt;-BCp^2&lt;/math&gt; || &lt;math&gt;+DAp&lt;/math&gt;  || &lt;math&gt;&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (9)&lt;/math&gt;
|-
| Simplify &lt;math&gt;(9)&lt;/math&gt; || &lt;math&gt;&lt;/math&gt;||&lt;math&gt;Dp^2&lt;/math&gt; || &lt;math&gt;-BCp&lt;/math&gt;  || &lt;math&gt;+DA&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (10)&lt;/math&gt;
|-
| &lt;math&gt;(6)*A&lt;/math&gt; || || &lt;math&gt;ACp^2&lt;/math&gt;|| &lt;math&gt;&lt;/math&gt;  || &lt;math&gt;+AD&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (11)&lt;/math&gt;
|-
| &lt;math&gt;(10)-(11)&lt;/math&gt; || &lt;math&gt;&lt;/math&gt;||&lt;math&gt;Dp^2-ACp^2&lt;/math&gt; || &lt;math&gt;-BCp&lt;/math&gt;  || &lt;math&gt;&lt;/math&gt;|| &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (12)&lt;/math&gt;
|-
| Simplify &lt;math&gt;(12)&lt;/math&gt; || &lt;math&gt;&lt;/math&gt;|| || &lt;math&gt;Dp-ACp&lt;/math&gt; || &lt;math&gt;-BC&lt;/math&gt;  || &lt;math&gt;= 0&lt;/math&gt; || &lt;math&gt;\dots (13)&lt;/math&gt;
|}

From &lt;math&gt;(13):\ p = \frac{BC}{D - AC} = \frac{aC}{AC - D} = \frac{-aC}{9Q + D} = \frac{aC}{b - 8Q}&lt;/math&gt;
{{RoundBoxBottom}}

==An Example==

Calculate cube roots of complex number &lt;math&gt;W = 39582 + 3799i.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:1219cubic01.png|thumb|400px|'''Graph of &lt;math&gt;f(Q)&lt;/math&gt; shown as graph of &lt;math&gt;f(x)&lt;/math&gt; and showing three values of &lt;math&gt;Q: Q_1, Q_2, Q_3&lt;/math&gt;.'''
&lt;/br&gt;
&lt;math&gt;Y&lt;/math&gt; axis compressed for clarity.
]]

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

a,b = 39582,3799

s = 64
t = 48*b
u = -(15*b**2 + 27*a**2)
v = b**3

s,t,u,v
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
(64, 182352, -42518323563, 54828691399)
&lt;/syntaxhighlight&gt;
Calculate roots of cubic function:
&lt;math&gt;y = f (x) &lt;/math&gt;&lt;math&gt; = 64 x^3 &lt;/math&gt;&lt;math&gt; + 182352 x^2 &lt;/math&gt;&lt;math&gt; - 42518323563 x &lt;/math&gt;&lt;math&gt; + 54828691399 .&lt;/math&gt;

Three roots are: &lt;math&gt;Q_1, Q_2, Q_3 = -27239.53953801976, 1.2895380197588122, 24389&lt;/math&gt;
{{RoundBoxBottom}}

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

values_of_Q = Q1,Q2,Q3 = -27239.53953801976, 1.2895380197588122, 24389
values_of_q = [ q
             for Q in values_of_Q
             for r in [ abs(Q) ** (1/3) ]
             for q in [ (r,-r)[Q&lt;0] ]
            ]
s1 = 'values_of_q' ; print(s1, eval(s1))

for q,Q in zip(values_of_q, values_of_Q) :
    C = 3*q
    p = a*C/(b - 8*Q)
    w = p + q*1j
    s1 = 'w, w**3' ; print (s1, eval(s1))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
values_of_q (-30.08845726811989, 1.0884572681198943, 29)
w, w**3 ((-16.114736709748723 - 30.08845726811989j  ), (39582+3799j))
w, w**3 (( 34.11473670974874  +  1.0884572681198943j), (39582+3799j))
w, w**3 ((-18.0               + 29.0j               ), (39582+3799j))
&lt;/syntaxhighlight&gt;
Three cube roots of &lt;math&gt;W = 39582 + 3799i&lt;/math&gt; are:

&lt;math&gt;w_1 = (-16.114736709748723 - 30.08845726811989i  )&lt;/math&gt;

&lt;math&gt;w_2 = ( 34.11473670974874  +  1.0884572681198943i)&lt;/math&gt;

&lt;math&gt;w_3 = (-18.0               + 29.0i               )&lt;/math&gt;
=Method #4 (Vieta Reversed)=

See [https://en.wikiversity.org/wiki/Cubic_function#Vieta's_substitution Vieta's Substitution.]

The depressed cubic function is : &lt;math&gt;f(t) = t^3 + At + B = 0.&lt;/math&gt;

Let &lt;math&gt;A = -3C&lt;/math&gt; and let &lt;math&gt;t = w + \frac{C}{w} = \frac{w^2 + C}{w}.&lt;/math&gt;

Substitute for &lt;math&gt;A, t&lt;/math&gt; in the depressed function and result is:

&lt;math&gt;f(w) = w^6 + Bw^3 + C^3&lt;/math&gt; or

&lt;math&gt;f(W) = W^2 + BW + C^3&lt;/math&gt; where &lt;math&gt;W = w^3&lt;/math&gt; and &lt;math&gt;w = \sqrt[3]{W}&lt;/math&gt;.

From the quadratic formula: &lt;math&gt;W = \frac{-B \pm \sqrt{B^2 - 4C^3}}{2} = \frac{-B \pm \sqrt{4C^3 - B^2}\sqrt{-1}   }{2}&lt;/math&gt;

However, &lt;math&gt;W = a + bi.&lt;/math&gt;

Therefore: &lt;math&gt;a = \frac{-B}{2}&lt;/math&gt; and &lt;math&gt;b = \frac{ \sqrt{4C^3 - B^2} }{2}.&lt;/math&gt;

&lt;math&gt;B = -2a&lt;/math&gt; and

&lt;math&gt;2b = \sqrt{4C^3 - B^2} &lt;/math&gt;

&lt;math&gt;4b^2 = 4C^3 - B^2 &lt;/math&gt;

&lt;math&gt;4C^3 = 4b^2 + B^2 &lt;/math&gt;

&lt;math&gt;C^3 = b^2 + \frac{B^2}{4} = b^2 + a^2&lt;/math&gt;

&lt;math&gt;C = \sqrt[3]{C^3} &lt;/math&gt; and

&lt;math&gt;A = -3C. &lt;/math&gt;

&lt;math&gt;A,B &lt;/math&gt; of &lt;math&gt;f(t)&lt;/math&gt; have been defined with &lt;math&gt;C&lt;/math&gt; positive and &lt;math&gt;A&lt;/math&gt; negative.

Because &lt;math&gt;W&lt;/math&gt; has three complex roots, &lt;math&gt;f(t)&lt;/math&gt; must have three real roots.

Calculate one of the roots of &lt;math&gt;f(t).&lt;/math&gt;

&lt;math&gt;t = \frac{w^2 + C}{w}.&lt;/math&gt;

Therefore &lt;math&gt;wt = w^2 + C&lt;/math&gt; and

&lt;math&gt;w^2 - tw + C = 0.&lt;/math&gt;

From the quadratic formula : 

&lt;math&gt;w = \frac{t \pm \sqrt{t^2 - 4C}}{2} = \frac{t \pm \sqrt{4C - t^2}\sqrt{-1}   }{2}&lt;/math&gt;

&lt;math&gt;w = p + qi&lt;/math&gt;

Therefore: &lt;math&gt;p = \frac{t}{2}&lt;/math&gt; and &lt;math&gt;Q = \frac{4C - t^2  }{4} = \frac{4C - 4P  }{4} = C - \frac{t^2}{4} = C - P&lt;/math&gt;
where &lt;math&gt;P = p^2,\ Q = q^2.&lt;/math&gt;

&lt;math&gt;q = \sqrt{Q}&lt;/math&gt; but sign of this calculation of &lt;math&gt;q&lt;/math&gt; is ambiguous.

&lt;math&gt;q = \frac{b}{(3P - Q)}&lt;/math&gt;

and &lt;math&gt;(p + qi)^3 = a + bi.&lt;/math&gt;

==An Example==

Calculate cube roots of complex number &lt;math&gt;W = -39582 + 3799i.&lt;/math&gt;
{{RoundBoxTop|theme=2}}
[[File:1215depressed cubic01.png|thumb|400px|'''Graph of &lt;math&gt;f(t)&lt;/math&gt; shown as graph of &lt;math&gt;f(x)&lt;/math&gt; and showing three values of &lt;math&gt;t: t_1, t_2, t_3&lt;/math&gt;.'''
&lt;/br&gt;
&lt;math&gt;Y&lt;/math&gt; axis compressed for clarity.
&lt;/br&gt;
&lt;math&gt;A,B = -3495.0, 79164.0&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;t_1 = -68.229473419497441512295943903670298\dots&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;t_2 = 32.229473419497441512295943903670298\dots&lt;/math&gt;
&lt;/br&gt;
&lt;math&gt;\ t_3 = 36&lt;/math&gt;
]]

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

import decimal
dD = decimal.Decimal
decimal.getcontext().prec = 22

ab = -39582,3799
a,b = [ dD(v) for v in ab ]
B = -2*a
C = (b**2 + a**2) ** (dD(1)/3)
A = -3*C

a,b,A,B,C = [ dD(str(float(v))) for v in (a,b,A,B,C) ]
print ( 'a,b = {}, {}'.format(a,b) )
print ( 'A,B,C = {}, {}, {}'.format(A,B,C) )
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
a,b = -39582.0, 3799.0
A,B,C = -3495.0, 79164.0, 1165.0
&lt;/syntaxhighlight&gt;
Calculate roots of cubic function:

&lt;math&gt;y = f (t) &lt;/math&gt;&lt;math&gt; = t^3 &lt;/math&gt;&lt;math&gt; - 3495 t &lt;/math&gt;&lt;math&gt; + 79164 .&lt;/math&gt;

Three roots are: &lt;math&gt;t_1,\ t_2,\ t_3 = -68.22947341949744\dots,\ 32.22947341949744\dots,\ 36.0&lt;/math&gt;
{{RoundBoxBottom}}
&lt;syntaxhighlight lang=python&gt;
# python code:
t1, t2, t3 = ('-68.229473419497441512295943903670298',
               '32.229473419497441512295943903670298', 36 )
values_of_t = [ dD(v) for v in (t1,t2,t3) ]

for t in values_of_t :
    p = t/2 ; P = p**2
    Q = C - P ; q = b/(3*P - Q)
    # Check results:
    print ()
    sx = 't' ; print (sx,'=', eval(sx))
    print ( 'p,q = {}, {}'.format(p,q) )
    ab = [ p*P - 3*p*Q, 3*P*q - q*Q ]
    a_,b_ = [ float(v) for v in ab ]
    sx = 'a_,b_' ; print (sx,'=', eval(sx))
&lt;/syntaxhighlight&gt;
&lt;syntaxhighlight&gt;
t = -68.229473419497441512295943903670298
p,q = -34.11473670974872075615, 1.088457268119895641747
a_,b_ = (-39582.0, 3799.0)
    
t = 32.229473419497441512295943903670298
p,q = 16.11473670974872075615, -30.08845726811989564176
a_,b_ = (-39582.0, 3799.0)
    
t = 36
p,q = 18, 29
a_,b_ = (-39582.0, 3799.0)
&lt;/syntaxhighlight&gt;
Three cube roots of &lt;math&gt;W = -39582 + 3799i&lt;/math&gt; are:

&lt;math&gt;w_1 = -34.11473670974872075615 + 1.088457268119895641747i&lt;/math&gt;

&lt;math&gt;w_2 = 16.11473670974872075615 - 30.08845726811989564176i&lt;/math&gt;

&lt;math&gt;w_3 = 18 + 29i&lt;/math&gt;</text>
      <sha1>anjwe4t7wqn5nrqi7b8aav0nhf3ndyk</sha1>
    </revision>
  </page>
  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
    <id>285380</id>
    <revision>
      <id>2815042</id>
      <parentid>2814866</parentid>
      <timestamp>2026-06-10T14:18:51Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2815042</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="i1rbmeuznsy3rm1h2gefz98pedv6wc4" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

=== Handling Repetition ===
* Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]])
* Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]])

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260610.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming language]]

&lt;/br&gt;</text>
      <sha1>i1rbmeuznsy3rm1h2gefz98pedv6wc4</sha1>
    </revision>
  </page>
  <page>
    <title>African Arthropods/Insects</title>
    <ns>0</ns>
    <id>287103</id>
    <revision>
      <id>2815237</id>
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      <timestamp>2026-06-11T11:32:30Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>Taxonomic sequence</comment>
      <origin>2815237</origin>
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      <text bytes="6392" sha1="h547a8ttu4j91207b8zyphgd25v76fe" xml:space="preserve">There are 27 orders of [[w:insect|insects]]. Scientific estimates have been made of the number of species in the different orders, but how common are they in terms of numbers of organisms or biomass? This is more difficult to measure, and there are many scientists asking this question, especially now that there is evidence of a [[w:Decline_in_insect_populations|world-wide decline in insect biomass]]. See also: [https://www.reuters.com/graphics/GLOBAL-ENVIRONMENT/INSECT-APOCALYPSE/egpbykdxjvq/ ''The collapse of insects: The most diverse group of organisms on the planet are in trouble, with recent research suggesting insect populations are declining at an unprecedented rate.'']

From an informal learning point of view, however, the number of records (in [https://www.inaturalist.org/home iNaturalist]) of organisms from each group gives an indication of how often people with cameras interact with that group. The table below gives a breakdown of the numbers of African records on iNaturalist for the different orders of insects. The table is interactive - one can sort the table so that it reflects the taxonomic order (based on the Wikipedia page for [[w:Insects|insects]]) by clicking on the column heading; the first two columns can be sorted alphabetically. 

[https://www.iziko.org.za/ Iziko museum] has a [http://www.biodiversityexplorer.info/insects/index.htm table of insect Orders] on their [http://www.biodiversityexplorer.info/index.htm biodiversity explorer site] for southern Africa

{| class=&quot;wikitable sortable&quot;
|+ 	Insect orders, showing the number of African records on iNaturalist on 25 July 2022	&lt;br&gt;

|-
! Order !! Common names !! Number of iNaturalist records !! Taxonomic sequence !! Images
|-
| [[African Arthropods/Lepidoptera|Lepidoptera]] || Butterflies and Moths || 226821 || 23 Amphiesmenoptera || [[File:African_Monarch_%28Danaus_chrysippus_aegyptius%29_%2817389277322%29.jpg|thumb]]
|-
| [[w:Coleoptera|Coleoptera]] || Beetles|| 81584 || 19 Coleopterida || [[File:Ground_Beetle_(Cypholoba_graphipteroides)_(11967049234).jpg|thumb]]
|-
| [[African_Arthropods/Hymenoptera|Hymenoptera]] || Ants, Bees, Wasps, and Sawflies || 55752 || 17 Hymenopterida || [[File:Polistes fastidiotus, a, Pretoria.jpg|thumb]]
|-
| [[w:Hemiptera|Hemiptera]] || True Bugs, Hoppers, Aphids, and Allies|| 41874 || 15 Acercaria || [[File:Giant Assassin Bug (Platymeris guttatipennis) (11838791835).jpg|thumb]]
|-
| [[African_Arthropods/Diptera|Diptera]] || Flies	|| 38453 || 25 Antliophora || [[File:Red-headed Fly (Bromophila caffra) on elephant dung (13943436532).jpg|thumb]]
|-
| [[w:Odonata|Odonata]] || Dragonflies and Damselflies || 36623 || 3 Odonatoptera || [[File:C pictus AManson 002992-2.jpg|thumb]]
|-
| Orthoptera || Grasshoppers, Crickets, and Katydids || 34887 || 8 Orthoptera || [[File:Phymateus morbillosus (Pyrgomorphidae) (4761619550).jpg|thumb]]
|-
| Mantodea || Mantises || 10403 || 9 Dictyoptera || [[File:Pseudocreobotra wahlbergii, nimf, a, Krantzkloof NR.jpg|thumb]]
|-
| Blattodea || Cockroaches and Termites || 9138 || 10 Dictyoptera || [[File:Black Cockroach (Deropeltis sp.) (11691293215).jpg|thumb]]
|-
| Neuroptera || Antlions, Lacewings, and Allies || 5318 || 21 Neuropterida || [[File:Micromus africanus 009733-1.jpg|thumb]]
|-
| Phasmida || Stick Insects || 1845 || 12 Eukinolabia || [[File:Giant_Stick_Insect_(Bactrododema_tiaratum)_(16917521442).jpg|thumb]]
|-
| Dermaptera || Earwigs || 756 || 6 Haplocercata || [[File:Earwig Walking.jpg|thumb]]
|-
| Ephemeroptera  || Mayflies || 719 || 4 Panephemeroptera || [[File:Ephemeroptera landscape orientation.jpg|thumb]]
|-
| Thysanoptera || Thrips || 477 || 16 Acercaria || [[File:Thrips Pietermaritzburg 2021 01 17.jpg|thumb]]
|-
| Zygentoma || Silverfishes || 451 || 2 Basal || [[File:Silverfish 2007-2.jpg|thumb]]
|-
| Trichoptera || Caddisflies || 425 || 24 Amphiesmenoptera || [[File:Genus Chimarra Little Black Caddisfly iNaturalist122139766.jpg|thumb]]
|-
| Psocodea || Barklice, Booklice, and Parasitic Lice || 326|| 14 Acercaria || [[File:Psocidae Common Barklice Psocodea iNat 29840951.jpg|thumb]]
|-
| Archaeognatha || Bristletails || 109 || 1 Basal || [[File:Rock Bristletail Taylor 2019 iNat 37124097.jpg|thumb]]
|-
| Embiidina || Webspinners || 100 || 13 Eukinolabia || [[File:2018 07 04 Haploembia solieri w.jpg|thumb]]
|-
| Plecoptera || Stoneflies || 60 || 7 Plecoptera || [[File:Stonefly Uys 2013 iNat 10892215.jpg|thumb]]
|-
| Megaloptera || Alderflies, Dobsonflies, and Fishflies || 41 || 22 Neuropterida || [[File:Dobson fly Andrew Deacon 2014 iNat 11059823.jpg|thumb]]
|-
| Mecoptera || Hangingflies, Scorpionflies, and Allies || 40 || 26 Antliophora || [[File:2016 01 31 12 29 Bittacus kimminsi.jpg|thumb]]
|-
| Siphonaptera || Fleas || 29 || 27 Antliophora || [[File:NHMUK010177296 A squirrel flea - Ceratophyllus Monopsyllus sciurorum (Schrank, 1803).jpg|thumb]]
|-
| Notoptera || Heelwalkers and Ice Crawlers || 16 || 11 Notoptera || [[File:Mantophasma zephyra Zompro et al 2002.jpg|thumb]]
|-
| Strepsiptera || Twisted-wing Insects || 7 || 18 Coleopterida || [[File:Family Stylopidae Manson 2019 Ngwenya.jpg|thumb]]
|-
| Raphidioptera || Snakeflies || 1 || 20 Neuropterida || [[File:Unidentified Inocelliidae Piazzo 02.jpg|thumb]]
|-
| Zoraptera || Angel Insects || 0 || 5 Haplocercata || [[File:Zorotypus from Los Bancos, Pichincha, Ecuador.jpg|thumb]]
|-
|}
Order [[w:Raphidioptera|Raphidioptera]] (snakeflies) is rare in Africa; Snakeflies are probably limited to a few countries on the [[w:Mediterranean_Sea|Mediterranean Sea]]. iNaturalist has only one African record; that is from Morocco.&lt;br&gt;
Order [[w:Zoraptera|Zoraptera]] (angel insects) are widespread worldwide, but are rarely recorded. There are no African records of angel insects on iNaturalist, but GBIF shows a record from [[w:Ethiopia|Ethiopia]].&lt;br&gt;

==Mapping Insect biodiversity in Africa==
The Biodiversity and Development Institute (BDI) and The FitzPatrick Institute of African Ornithology run a citizen science programme called the [https://vmus.adu.org.za/ Virtual Museum] - this includes these projects:
&lt;gallery&gt;
DungBeetleMAP logo.png|Atlas of Dung Beetles in southern Africa
LacewingMAP logo.png|Atlas of African Neuroptera and Megaloptera
Lepimap logo.png|Atlas of African Lepidoptera
Odonata logo.png|Odonata Atlas of Africa
&lt;/gallery&gt;

[[Category:African Arthropods]]
[[Category:Insects]]</text>
      <sha1>h547a8ttu4j91207b8zyphgd25v76fe</sha1>
    </revision>
    <revision>
      <id>2815238</id>
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        <username>Alandmanson</username>
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      <text bytes="6415" sha1="ek8su5g0nvg94q4th20hlpiwx5jhsvy" xml:space="preserve">There are 27 orders of [[w:insect|insects]]. Scientific estimates have been made of the number of species in the different orders, but how common are they in terms of numbers of organisms or biomass? This is more difficult to measure, and there are many scientists asking this question, especially now that there is evidence of a [[w:Decline_in_insect_populations|world-wide decline in insect biomass]]. See also: [https://www.reuters.com/graphics/GLOBAL-ENVIRONMENT/INSECT-APOCALYPSE/egpbykdxjvq/ ''The collapse of insects: The most diverse group of organisms on the planet are in trouble, with recent research suggesting insect populations are declining at an unprecedented rate.'']

From an informal learning point of view, however, the number of records (in [https://www.inaturalist.org/home iNaturalist]) of organisms from each group gives an indication of how often people with cameras interact with that group. The table below gives a breakdown of the numbers of African records on iNaturalist for the different orders of insects. The table is interactive - one can sort the table so that it reflects the taxonomic order (based on the Wikipedia page for [[w:Insects|insects]]) by clicking on the column heading; the first two columns can be sorted alphabetically. 

[https://www.iziko.org.za/ Iziko museum] has a [http://www.biodiversityexplorer.info/insects/index.htm table of insect Orders] on their [http://www.biodiversityexplorer.info/index.htm biodiversity explorer site] for southern Africa

{| class=&quot;wikitable sortable&quot;
|+ 	Insect orders, showing the number of African records on iNaturalist on 25 July 2022	&lt;br&gt;

|-
! Order !! Common names !! Number of iNaturalist records !! [[w:Taxonomic sequence|Taxonomic sequence !! Images
|-
| [[African Arthropods/Lepidoptera|Lepidoptera]] || Butterflies and Moths || 226821 || 23 Amphiesmenoptera || [[File:African_Monarch_%28Danaus_chrysippus_aegyptius%29_%2817389277322%29.jpg|thumb]]
|-
| [[w:Coleoptera|Coleoptera]] || Beetles|| 81584 || 19 Coleopterida || [[File:Ground_Beetle_(Cypholoba_graphipteroides)_(11967049234).jpg|thumb]]
|-
| [[African_Arthropods/Hymenoptera|Hymenoptera]] || Ants, Bees, Wasps, and Sawflies || 55752 || 17 Hymenopterida || [[File:Polistes fastidiotus, a, Pretoria.jpg|thumb]]
|-
| [[w:Hemiptera|Hemiptera]] || True Bugs, Hoppers, Aphids, and Allies|| 41874 || 15 Acercaria || [[File:Giant Assassin Bug (Platymeris guttatipennis) (11838791835).jpg|thumb]]
|-
| [[African_Arthropods/Diptera|Diptera]] || Flies	|| 38453 || 25 Antliophora || [[File:Red-headed Fly (Bromophila caffra) on elephant dung (13943436532).jpg|thumb]]
|-
| [[w:Odonata|Odonata]] || Dragonflies and Damselflies || 36623 || 3 Odonatoptera || [[File:C pictus AManson 002992-2.jpg|thumb]]
|-
| Orthoptera || Grasshoppers, Crickets, and Katydids || 34887 || 8 Orthoptera || [[File:Phymateus morbillosus (Pyrgomorphidae) (4761619550).jpg|thumb]]
|-
| Mantodea || Mantises || 10403 || 9 Dictyoptera || [[File:Pseudocreobotra wahlbergii, nimf, a, Krantzkloof NR.jpg|thumb]]
|-
| Blattodea || Cockroaches and Termites || 9138 || 10 Dictyoptera || [[File:Black Cockroach (Deropeltis sp.) (11691293215).jpg|thumb]]
|-
| Neuroptera || Antlions, Lacewings, and Allies || 5318 || 21 Neuropterida || [[File:Micromus africanus 009733-1.jpg|thumb]]
|-
| Phasmida || Stick Insects || 1845 || 12 Eukinolabia || [[File:Giant_Stick_Insect_(Bactrododema_tiaratum)_(16917521442).jpg|thumb]]
|-
| Dermaptera || Earwigs || 756 || 6 Haplocercata || [[File:Earwig Walking.jpg|thumb]]
|-
| Ephemeroptera  || Mayflies || 719 || 4 Panephemeroptera || [[File:Ephemeroptera landscape orientation.jpg|thumb]]
|-
| Thysanoptera || Thrips || 477 || 16 Acercaria || [[File:Thrips Pietermaritzburg 2021 01 17.jpg|thumb]]
|-
| Zygentoma || Silverfishes || 451 || 2 Basal || [[File:Silverfish 2007-2.jpg|thumb]]
|-
| Trichoptera || Caddisflies || 425 || 24 Amphiesmenoptera || [[File:Genus Chimarra Little Black Caddisfly iNaturalist122139766.jpg|thumb]]
|-
| Psocodea || Barklice, Booklice, and Parasitic Lice || 326|| 14 Acercaria || [[File:Psocidae Common Barklice Psocodea iNat 29840951.jpg|thumb]]
|-
| Archaeognatha || Bristletails || 109 || 1 Basal || [[File:Rock Bristletail Taylor 2019 iNat 37124097.jpg|thumb]]
|-
| Embiidina || Webspinners || 100 || 13 Eukinolabia || [[File:2018 07 04 Haploembia solieri w.jpg|thumb]]
|-
| Plecoptera || Stoneflies || 60 || 7 Plecoptera || [[File:Stonefly Uys 2013 iNat 10892215.jpg|thumb]]
|-
| Megaloptera || Alderflies, Dobsonflies, and Fishflies || 41 || 22 Neuropterida || [[File:Dobson fly Andrew Deacon 2014 iNat 11059823.jpg|thumb]]
|-
| Mecoptera || Hangingflies, Scorpionflies, and Allies || 40 || 26 Antliophora || [[File:2016 01 31 12 29 Bittacus kimminsi.jpg|thumb]]
|-
| Siphonaptera || Fleas || 29 || 27 Antliophora || [[File:NHMUK010177296 A squirrel flea - Ceratophyllus Monopsyllus sciurorum (Schrank, 1803).jpg|thumb]]
|-
| Notoptera || Heelwalkers and Ice Crawlers || 16 || 11 Notoptera || [[File:Mantophasma zephyra Zompro et al 2002.jpg|thumb]]
|-
| Strepsiptera || Twisted-wing Insects || 7 || 18 Coleopterida || [[File:Family Stylopidae Manson 2019 Ngwenya.jpg|thumb]]
|-
| Raphidioptera || Snakeflies || 1 || 20 Neuropterida || [[File:Unidentified Inocelliidae Piazzo 02.jpg|thumb]]
|-
| Zoraptera || Angel Insects || 0 || 5 Haplocercata || [[File:Zorotypus from Los Bancos, Pichincha, Ecuador.jpg|thumb]]
|-
|}
Order [[w:Raphidioptera|Raphidioptera]] (snakeflies) is rare in Africa; Snakeflies are probably limited to a few countries on the [[w:Mediterranean_Sea|Mediterranean Sea]]. iNaturalist has only one African record; that is from Morocco.&lt;br&gt;
Order [[w:Zoraptera|Zoraptera]] (angel insects) are widespread worldwide, but are rarely recorded. There are no African records of angel insects on iNaturalist, but GBIF shows a record from [[w:Ethiopia|Ethiopia]].&lt;br&gt;

==Mapping Insect biodiversity in Africa==
The Biodiversity and Development Institute (BDI) and The FitzPatrick Institute of African Ornithology run a citizen science programme called the [https://vmus.adu.org.za/ Virtual Museum] - this includes these projects:
&lt;gallery&gt;
DungBeetleMAP logo.png|Atlas of Dung Beetles in southern Africa
LacewingMAP logo.png|Atlas of African Neuroptera and Megaloptera
Lepimap logo.png|Atlas of African Lepidoptera
Odonata logo.png|Odonata Atlas of Africa
&lt;/gallery&gt;

[[Category:African Arthropods]]
[[Category:Insects]]</text>
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      <text bytes="6417" sha1="3fokpuyunbf4xe7pdxx0z3mks9cmiqn" xml:space="preserve">There are 27 orders of [[w:insect|insects]]. Scientific estimates have been made of the number of species in the different orders, but how common are they in terms of numbers of organisms or biomass? This is more difficult to measure, and there are many scientists asking this question, especially now that there is evidence of a [[w:Decline_in_insect_populations|world-wide decline in insect biomass]]. See also: [https://www.reuters.com/graphics/GLOBAL-ENVIRONMENT/INSECT-APOCALYPSE/egpbykdxjvq/ ''The collapse of insects: The most diverse group of organisms on the planet are in trouble, with recent research suggesting insect populations are declining at an unprecedented rate.'']

From an informal learning point of view, however, the number of records (in [https://www.inaturalist.org/home iNaturalist]) of organisms from each group gives an indication of how often people with cameras interact with that group. The table below gives a breakdown of the numbers of African records on iNaturalist for the different orders of insects. The table is interactive - one can sort the table so that it reflects the taxonomic order (based on the Wikipedia page for [[w:Insects|insects]]) by clicking on the column heading; the first two columns can be sorted alphabetically. 

[https://www.iziko.org.za/ Iziko museum] has a [http://www.biodiversityexplorer.info/insects/index.htm table of insect Orders] on their [http://www.biodiversityexplorer.info/index.htm biodiversity explorer site] for southern Africa

{| class=&quot;wikitable sortable&quot;
|+ 	Insect orders, showing the number of African records on iNaturalist on 25 July 2022	&lt;br&gt;

|-
! Order !! Common names !! Number of iNaturalist records !! [[w:Taxonomic sequence|Taxonomic sequence]] !! Images
|-
| [[African Arthropods/Lepidoptera|Lepidoptera]] || Butterflies and Moths || 226821 || 23 Amphiesmenoptera || [[File:African_Monarch_%28Danaus_chrysippus_aegyptius%29_%2817389277322%29.jpg|thumb]]
|-
| [[w:Coleoptera|Coleoptera]] || Beetles|| 81584 || 19 Coleopterida || [[File:Ground_Beetle_(Cypholoba_graphipteroides)_(11967049234).jpg|thumb]]
|-
| [[African_Arthropods/Hymenoptera|Hymenoptera]] || Ants, Bees, Wasps, and Sawflies || 55752 || 17 Hymenopterida || [[File:Polistes fastidiotus, a, Pretoria.jpg|thumb]]
|-
| [[w:Hemiptera|Hemiptera]] || True Bugs, Hoppers, Aphids, and Allies|| 41874 || 15 Acercaria || [[File:Giant Assassin Bug (Platymeris guttatipennis) (11838791835).jpg|thumb]]
|-
| [[African_Arthropods/Diptera|Diptera]] || Flies	|| 38453 || 25 Antliophora || [[File:Red-headed Fly (Bromophila caffra) on elephant dung (13943436532).jpg|thumb]]
|-
| [[w:Odonata|Odonata]] || Dragonflies and Damselflies || 36623 || 3 Odonatoptera || [[File:C pictus AManson 002992-2.jpg|thumb]]
|-
| Orthoptera || Grasshoppers, Crickets, and Katydids || 34887 || 8 Orthoptera || [[File:Phymateus morbillosus (Pyrgomorphidae) (4761619550).jpg|thumb]]
|-
| Mantodea || Mantises || 10403 || 9 Dictyoptera || [[File:Pseudocreobotra wahlbergii, nimf, a, Krantzkloof NR.jpg|thumb]]
|-
| Blattodea || Cockroaches and Termites || 9138 || 10 Dictyoptera || [[File:Black Cockroach (Deropeltis sp.) (11691293215).jpg|thumb]]
|-
| Neuroptera || Antlions, Lacewings, and Allies || 5318 || 21 Neuropterida || [[File:Micromus africanus 009733-1.jpg|thumb]]
|-
| Phasmida || Stick Insects || 1845 || 12 Eukinolabia || [[File:Giant_Stick_Insect_(Bactrododema_tiaratum)_(16917521442).jpg|thumb]]
|-
| Dermaptera || Earwigs || 756 || 6 Haplocercata || [[File:Earwig Walking.jpg|thumb]]
|-
| Ephemeroptera  || Mayflies || 719 || 4 Panephemeroptera || [[File:Ephemeroptera landscape orientation.jpg|thumb]]
|-
| Thysanoptera || Thrips || 477 || 16 Acercaria || [[File:Thrips Pietermaritzburg 2021 01 17.jpg|thumb]]
|-
| Zygentoma || Silverfishes || 451 || 2 Basal || [[File:Silverfish 2007-2.jpg|thumb]]
|-
| Trichoptera || Caddisflies || 425 || 24 Amphiesmenoptera || [[File:Genus Chimarra Little Black Caddisfly iNaturalist122139766.jpg|thumb]]
|-
| Psocodea || Barklice, Booklice, and Parasitic Lice || 326|| 14 Acercaria || [[File:Psocidae Common Barklice Psocodea iNat 29840951.jpg|thumb]]
|-
| Archaeognatha || Bristletails || 109 || 1 Basal || [[File:Rock Bristletail Taylor 2019 iNat 37124097.jpg|thumb]]
|-
| Embiidina || Webspinners || 100 || 13 Eukinolabia || [[File:2018 07 04 Haploembia solieri w.jpg|thumb]]
|-
| Plecoptera || Stoneflies || 60 || 7 Plecoptera || [[File:Stonefly Uys 2013 iNat 10892215.jpg|thumb]]
|-
| Megaloptera || Alderflies, Dobsonflies, and Fishflies || 41 || 22 Neuropterida || [[File:Dobson fly Andrew Deacon 2014 iNat 11059823.jpg|thumb]]
|-
| Mecoptera || Hangingflies, Scorpionflies, and Allies || 40 || 26 Antliophora || [[File:2016 01 31 12 29 Bittacus kimminsi.jpg|thumb]]
|-
| Siphonaptera || Fleas || 29 || 27 Antliophora || [[File:NHMUK010177296 A squirrel flea - Ceratophyllus Monopsyllus sciurorum (Schrank, 1803).jpg|thumb]]
|-
| Notoptera || Heelwalkers and Ice Crawlers || 16 || 11 Notoptera || [[File:Mantophasma zephyra Zompro et al 2002.jpg|thumb]]
|-
| Strepsiptera || Twisted-wing Insects || 7 || 18 Coleopterida || [[File:Family Stylopidae Manson 2019 Ngwenya.jpg|thumb]]
|-
| Raphidioptera || Snakeflies || 1 || 20 Neuropterida || [[File:Unidentified Inocelliidae Piazzo 02.jpg|thumb]]
|-
| Zoraptera || Angel Insects || 0 || 5 Haplocercata || [[File:Zorotypus from Los Bancos, Pichincha, Ecuador.jpg|thumb]]
|-
|}
Order [[w:Raphidioptera|Raphidioptera]] (snakeflies) is rare in Africa; Snakeflies are probably limited to a few countries on the [[w:Mediterranean_Sea|Mediterranean Sea]]. iNaturalist has only one African record; that is from Morocco.&lt;br&gt;
Order [[w:Zoraptera|Zoraptera]] (angel insects) are widespread worldwide, but are rarely recorded. There are no African records of angel insects on iNaturalist, but GBIF shows a record from [[w:Ethiopia|Ethiopia]].&lt;br&gt;

==Mapping Insect biodiversity in Africa==
The Biodiversity and Development Institute (BDI) and The FitzPatrick Institute of African Ornithology run a citizen science programme called the [https://vmus.adu.org.za/ Virtual Museum] - this includes these projects:
&lt;gallery&gt;
DungBeetleMAP logo.png|Atlas of Dung Beetles in southern Africa
LacewingMAP logo.png|Atlas of African Neuroptera and Megaloptera
Lepimap logo.png|Atlas of African Lepidoptera
Odonata logo.png|Odonata Atlas of Africa
&lt;/gallery&gt;

[[Category:African Arthropods]]
[[Category:Insects]]</text>
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      <text bytes="6473" sha1="86hydslypqarz2ta8w5tts47zil3dxc" xml:space="preserve">There are 27 orders of [[w:insect|insects]]. Scientific estimates have been made of the number of species in the different orders, but how common are they in terms of numbers of organisms or biomass? This is more difficult to measure, and there are many scientists asking this question, especially now that there is evidence of a [[w:Decline_in_insect_populations|world-wide decline in insect biomass]]. See also: [https://www.reuters.com/graphics/GLOBAL-ENVIRONMENT/INSECT-APOCALYPSE/egpbykdxjvq/ ''The collapse of insects: The most diverse group of organisms on the planet are in trouble, with recent research suggesting insect populations are declining at an unprecedented rate.'']

From an informal learning point of view, however, the number of records (in [https://www.inaturalist.org/home iNaturalist]) of organisms from each group gives an indication of how often people with cameras interact with that group. The table below gives a breakdown of the numbers of African records on iNaturalist for the different orders of insects. The table is interactive - one can sort the table so that it reflects the taxonomic order (based on the Wikipedia page for [[w:Insects|insects]]) by clicking on the column heading; the first two columns can be sorted alphabetically. 

[https://www.iziko.org.za/ Iziko museum] has a [http://www.biodiversityexplorer.info/insects/index.htm table of insect Orders] on their [http://www.biodiversityexplorer.info/index.htm biodiversity explorer site] for southern Africa

{| class=&quot;wikitable sortable&quot;
|+ 	Insect orders, showing the number of African records on iNaturalist on 25 July 2022	&lt;br&gt;

|-
! Order !! Common names !! Number of iNaturalist records !! [[w:Taxonomic sequence|Taxonomic sequence]] !! Images
|-
| [[African Arthropods/Lepidoptera|Lepidoptera]] || Butterflies and Moths || 226821 || 23 Amphiesmenoptera || [[File:African_Monarch_%28Danaus_chrysippus_aegyptius%29_%2817389277322%29.jpg|thumb]]
|-
| [[w:Coleoptera|Coleoptera]] || Beetles|| 81584 || 19 Coleopterida || [[File:Ground_Beetle_(Cypholoba_graphipteroides)_(11967049234).jpg|thumb]]
|-
| [[African_Arthropods/Hymenoptera|Hymenoptera]] || Ants, Bees, Wasps, and Sawflies || 55752 || 17 Hymenopterida || [[File:Polistes fastidiotus, a, Pretoria.jpg|thumb]]
|-
| [[w:Hemiptera|Hemiptera]] || True Bugs, Hoppers, Aphids, and Allies|| 41874 || 15 Acercaria || [[File:Giant Assassin Bug (Platymeris guttatipennis) (11838791835).jpg|thumb]]
|-
| [[African_Arthropods/Diptera|Diptera]] || Flies	|| 38453 || 25 Antliophora || [[File:Red-headed Fly (Bromophila caffra) on elephant dung (13943436532).jpg|thumb]]
|-
| [[w:Odonata|Odonata]] || Dragonflies and Damselflies || 36623 || 3 Odonatoptera || [[File:C pictus AManson 002992-2.jpg|thumb]]
|-
| Orthoptera || Grasshoppers, Crickets, and Katydids || 34887 || 8 Orthoptera || [[File:Phymateus morbillosus (Pyrgomorphidae) (4761619550).jpg|thumb]]
|-
| Mantodea || Mantises || 10403 || 9 Dictyoptera || [[File:Pseudocreobotra wahlbergii, nimf, a, Krantzkloof NR.jpg|thumb]]
|-
| Blattodea || Cockroaches and Termites || 9138 || 10 Dictyoptera || [[File:Black Cockroach (Deropeltis sp.) (11691293215).jpg|thumb]]
|-
| Neuroptera || Antlions, Lacewings, and Allies || 5318 || 21 Neuropterida || [[File:Micromus africanus 009733-1.jpg|thumb]]
|-
| Phasmida || Stick Insects || 1845 || 12 Eukinolabia || [[File:Giant_Stick_Insect_(Bactrododema_tiaratum)_(16917521442).jpg|thumb]]
|-
| Dermaptera || Earwigs || 756 || 6 Haplocercata || [[File:Earwig Walking.jpg|thumb]]
|-
| Ephemeroptera  || Mayflies || 719 || 4 Panephemeroptera || [[File:Ephemeroptera landscape orientation.jpg|thumb]]
|-
| Thysanoptera || Thrips || 477 || 16 Acercaria || [[File:Thrips Pietermaritzburg 2021 01 17.jpg|thumb]]
|-
| Zygentoma || Silverfishes || 451 || 2 [[w:Basal (phylogenetics)|Basal]] || [[File:Silverfish 2007-2.jpg|thumb]]
|-
| Trichoptera || Caddisflies || 425 || 24 Amphiesmenoptera || [[File:Genus Chimarra Little Black Caddisfly iNaturalist122139766.jpg|thumb]]
|-
| Psocodea || Barklice, Booklice, and Parasitic Lice || 326|| 14 Acercaria || [[File:Psocidae Common Barklice Psocodea iNat 29840951.jpg|thumb]]
|-
| Archaeognatha || Bristletails || 109 || 1 [[w:Basal (phylogenetics)|Basal]] || [[File:Rock Bristletail Taylor 2019 iNat 37124097.jpg|thumb]]
|-
| Embiidina || Webspinners || 100 || 13 Eukinolabia || [[File:2018 07 04 Haploembia solieri w.jpg|thumb]]
|-
| Plecoptera || Stoneflies || 60 || 7 Plecoptera || [[File:Stonefly Uys 2013 iNat 10892215.jpg|thumb]]
|-
| Megaloptera || Alderflies, Dobsonflies, and Fishflies || 41 || 22 Neuropterida || [[File:Dobson fly Andrew Deacon 2014 iNat 11059823.jpg|thumb]]
|-
| Mecoptera || Hangingflies, Scorpionflies, and Allies || 40 || 26 Antliophora || [[File:2016 01 31 12 29 Bittacus kimminsi.jpg|thumb]]
|-
| Siphonaptera || Fleas || 29 || 27 Antliophora || [[File:NHMUK010177296 A squirrel flea - Ceratophyllus Monopsyllus sciurorum (Schrank, 1803).jpg|thumb]]
|-
| Notoptera || Heelwalkers and Ice Crawlers || 16 || 11 Notoptera || [[File:Mantophasma zephyra Zompro et al 2002.jpg|thumb]]
|-
| Strepsiptera || Twisted-wing Insects || 7 || 18 Coleopterida || [[File:Family Stylopidae Manson 2019 Ngwenya.jpg|thumb]]
|-
| Raphidioptera || Snakeflies || 1 || 20 Neuropterida || [[File:Unidentified Inocelliidae Piazzo 02.jpg|thumb]]
|-
| Zoraptera || Angel Insects || 0 || 5 Haplocercata || [[File:Zorotypus from Los Bancos, Pichincha, Ecuador.jpg|thumb]]
|-
|}
Order [[w:Raphidioptera|Raphidioptera]] (snakeflies) is rare in Africa; Snakeflies are probably limited to a few countries on the [[w:Mediterranean_Sea|Mediterranean Sea]]. iNaturalist has only one African record; that is from Morocco.&lt;br&gt;
Order [[w:Zoraptera|Zoraptera]] (angel insects) are widespread worldwide, but are rarely recorded. There are no African records of angel insects on iNaturalist, but GBIF shows a record from [[w:Ethiopia|Ethiopia]].&lt;br&gt;

==Mapping Insect biodiversity in Africa==
The Biodiversity and Development Institute (BDI) and The FitzPatrick Institute of African Ornithology run a citizen science programme called the [https://vmus.adu.org.za/ Virtual Museum] - this includes these projects:
&lt;gallery&gt;
DungBeetleMAP logo.png|Atlas of Dung Beetles in southern Africa
LacewingMAP logo.png|Atlas of African Neuroptera and Megaloptera
Lepimap logo.png|Atlas of African Lepidoptera
Odonata logo.png|Odonata Atlas of Africa
&lt;/gallery&gt;

[[Category:African Arthropods]]
[[Category:Insects]]</text>
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      <text bytes="1677" sha1="novvfcrkir89qlca46rykkmjhc81s9r" xml:space="preserve">{{title|Motivation and emotion book project overview}}
This &lt;noinclude&gt;[[Motivation and emotion/Book|project]]&lt;/noinclude&gt;&lt;includeonly&gt;project&lt;/includeonly&gt; has created over 1,800 online book chapters about how psychological science about motivation and emotion can be used to improve human lives. [[open academia|Open academic]] teaching principles are used to engage students in collaborative authorship to create [[Open Educational Resources|open educational resources]]. The work is published via [[Wikiversity]], an open wiki platform supported by the [[Wikimedia Foundation]]. Third year undergraduate psychology students develop the chapters as a learning and assessment exercise for the [[Motivation and emotion|motivation and emotion]] unit at the [[University of Canberra]], Australia.&lt;noinclude&gt;

{{RoundBoxTop}}
&lt;div align=&quot;center&quot;&gt;Book theme:&lt;/div&gt;
&lt;div align=&quot;center&quot;&gt;''How can we improve our motivational and emotional lives based on psychological science (theory and research)''?&lt;/div&gt;
{{RoundBoxBottom}}

Learn more about this project from an educational psychology perspective:
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki: An open education case study]] (Neill, 2024)
* [[User:Jtneill/Publications/Wikis provide a rich environment for collaborative open educational practices: Motivation and emotion case study|Wikis provide a rich environment for collaborative open educational practices]] (Neill, 2024)
&lt;!--
Also of interest may be the:&lt;br&gt;
[https://x.com/jtneill/status/1422496816709312513 2021 book chapter highlights Twitter thread].
--&gt;
[[Category:Motivation and emotion/Book]]&lt;/noinclude&gt;</text>
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  <page>
    <title>Motivation and emotion/Book/2023/Novelty-variety as a psychological need</title>
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      <text bytes="21902" sha1="eee5tq48u4bsuymej32qk2ckbdd8mnh" xml:space="preserve">{{title| Novelty-variety as a psychological need:&lt;br&gt;What is novelty-variety and what are its implications as a psychological need?}}
{{MECR3|1=https://youtu.be/ZHxIHBlWMSQ}}
__TOC__

==Overview==
[[File:-073 365- Joy (3858051663).jpg|thumb|356x356px|'''Figure 1'''. Children joyfully engaged in the novel experience of their first encounter with snow.]]

[[w:Novelty|Novelty]] is considered to be the [[w:cognitive process|cognitive process]] used to identify a stimulus previously unknown, and new (Siddique, 2017). Novelty has been found to reinforce learning and promote [[w:intrinsic motivation|intrinsic motivation]] (Siddique, 2017). [[w:Variety|Variety]], on the other hand, refers to the quality or state of being diverse and varied (Wentworth &amp; Witryol, 2012).

These two concepts are distinct but also intertwined. Variety is essential for a novel experience, however, not all variety is necessarily novel. That is, a novel experience will vary with respect to previous experiences, but an experience can vary from another prior experience without having to be novel (Gonzalez-Cutre, 2016). The word ‘[[w:novelty-variety|novelty-variety]]’ is often used together in order to refer to an experience that is new and different as well as diverse.

Novelty-variety must be explored in further depth to make a considerable claim for a potential fourth candidate to the pre-existing three [[w:self-determination theory|self-determination theory]] variables namely autonomy, relatedness and competence.

==Current state of the literature==

Before exploring novelty-variety and its classification as a basic [[w:psychological need|psychological need]], it is important to determine what basic psychological needs are. Ryan and Deci’s (2000) self-determination theory refers to three basic psychological needs all human beings require for optimal human functioning (Gonzalez-Cutre, 2016). These three needs are autonomy, competence and relatedness. According to Ryan and Desi (2000), these needs are [[w:innate|innate]] and therefore, are not learned or developed behaviour, and they are all required for an individual’s growth, integrity and well-being (Gonzalez-Cutre, 2016). An extensive amount of research has been conducted to show that these three psychological needs strongly promote [[w:wellbeing|wellbeing]] of individuals across [[w:cultures|cultures]], satisfy optimal life functions and positive growth. According to Baumeister and Leary (1995) and Ryan and Deci (2017), the six criteria that a need must meet in order to be a basic psychological need are as follows:

*Have strong association with [[w:psychological integrity|psychological integrity]], health and wellbeing with its frustration negatively associated with same and poor functioning;
*Have [[w:explicit|explicit]] specification of experiences and behaviours that lead to wellbeing;
*Be essential to the interpretation of the current [[w:empirical|empirical]] phenomena;
*Be a ‘growth need’ that works in [[w:interaction|interaction]] with the other three basic psychological needs as opposed to only when the other basic needs are thwarted;
*Be the [[w:precursor|precursor]] of the increase of intrinsic motivation and [[w:organismic|organismic]] integration; and
*Operate [[w:universally|universally]] for all people regardless of their age, culture and more. 

Self-determination theory consistently highlights the importance of novelty and unique challenges in increasing intrinsic motivation. However, earlier research prioritises autonomy and competence as the main drivers of intrinsic motivation and novelty-variety was often a factor discussed in relation to autonomy and competence. The current state of the literature shows that novelty-variety, as a basic psychological need, is a newly emerging area of research. Recent findings suggest that novelty-variety must be proposed as a separate [[w:construct|construct]] that promotes intrinsic motivation. Accordingly, in addition to the three psychological needs as explored in the self-determination theory, novelty-variety should be regarded as another psychological need that is necessary and important for human growth and [[w:satisfaction|satisfaction]] (Gonzalez-Cutre, 2016).

{{Robelbox|theme=1|title=Case study - Alex|iconwidth=48px}}&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
Alex is a 23 year old chemistry student with a unique motivation style. Traditional study methods bore them, but they thrive on challenge. To fuel engagement, Alex designs intricate study games, solving meta problems that relate to their study materials. The novelty of this approach not only sparks their curiosity but also drives them to delve deeper into their studies.
&lt;/div&gt;
{{Robelbox-close}}

==Novelty-variety: A case for acknowledgment==
The need for novelty-variety is found to be related to [[w:adaptive|adaptive]] social outcomes as when people experience novelty-variety, there is an increase in their intrinsic motivation and [[w:relatedness|relatedness]] (Gonzalez-Cutre, 2016). Research with respect to intrinsic motivation shows that there are four different factors that support an individual’s intrinsic motivation and overall wellbeing: interest, curiosity, sensation seeking and perceived variety (Gonzalez-Cutre, 2016). Interest, as found by Silvia (2008) suggests that intrinsic motivation is subject to an individual’s assessment of the novelty-complexity of an experience and the individual’s assessment of the [[w:comprehensibility|comprehensibility]] of the experience, [[w:curiosity|curiosity]] refers to an individual’s [[w:predisposition|predisposition]] to recognise and seek new experiences, sensation seeking, as developed by Zuckerman (1979) refers to the need for novel and varied and complex sensation and experiences and perceived variety establishes that varied and unexpected behaviour continuously promotes individuals’ wellbeing (Gonzalez-Cutre, 2016). In accordance with the above studies intrinsic motivation, which self-determination theory is larged based on, makes continuous reference to novelty and variety as a necessary and prominent construct of intrinsic motivation{{fact}}. Therefore, it is inevitable that novelty-variety is continuously proposed as a separate basic psychological need. 

For novelty-variety to be acknowledged as a basic psychological need, it must adhere to the [[w:framework|framework]] laid out in the basic needs theory posited by Baumeister and Leary (1995) and Ryan and Deci (2017). Various research and findings suggest that novelty-variety is able to meet the six criteria as follows, supporting the argument that novelty-variety should be considered a basic psychological need.

[[File:Deci and Ryan.jpg|thumb|300px|'''Figure 2'''. Richard Ryan and Edward Deci - founders of the self-determination theory]]

===Association with wellbeing and frustration negatively associated with poor functioning===
It is considered that novelty-variety exhibits similar properties to the other three basic psychological needs and is deemed necessary for [[w:life satisfaction|life satisfaction]] (Bhageri, 2019) and when absent, thwarts positive wellbeing. (Sheldon, 2001)

Novelty-variety has repeatedly shown in studies to improve and influence life satisfaction. Gonzalez-Cutre et al. (2020) highlights a benefit to novelty-variety in the physical education context whereby students have claimed to experience higher levels of intrinsic motivation as a product of slight increase in variance of physical activity, therefore, supports the satisfaction of the first [[w:criteria|criteria]] of a basic psychological need.

Additionally, novelty-variety is [[w:interrelated|interrelated]] with positive social outcomes whereby the experience of a novel activity is sought to be shared with another providing intrinsic motivation and relatedness complementing the already determined psychological needs in the self-determination theory. Hence, the satisfaction of novelty needs will satisfy the conditions for psychological integrity providing positive wellbeing (Gozalez-Cutre, 2016). In a learning perspective, with English learning Japanese students, the frustration of the variables within the self-determination theory as well as novelty-variety caused negative effects on wellbeing which further suggests that the absence of novelty-variety negatively influences people’s wellbeing, which, again supports the first criterion of a basic psychological need.

Therefore, the satisfaction of novelty-variety not only aids the other three basic psychological needs in promoting a positive overall wellbeing and optimal functioning, the frustration of novelty-variety also plays a strong role in adversely affecting the individual causing maladaptive life functions. 

===Explicit specification of experiences and behaviours===
According to Ryan and Deci (2017), it is important that definitions of novelty-variety must explicitly specify types of activities that will lead to well-being. It can be found in studies of novelty-variety that researchers do specifically mention types of [[w:activities|activities]] and actions that lead to well-being.

For example, in investigating the first criterion, González-Cutre et al. (2016) made specific [[w:theoretical|theoretical]] descriptions of experiences and behaviors with respect to well-being that indicate the need for novelty-variety{{example}}. With the first criterion found to be met by González-Cutre et al, these descriptions address the second criterion required{{vague}}.

===Essential to the interpretation of the current empirical phenomena===
Ryan and Deci (2017) requires all basic psychological needs to be a consistent [[w:mediator|mediator]] of social and personal factors and individuals’ motivations and psychosocial functioning.

Accordingly, González-Cutre et al. (2016) hypothesised that motivation would mediate effects of novelty on outcomes and therefore, satisfy the third criterion. In finding a positive correlation between the satisfaction of novelty-variety and motivation and well-being, González-Cutre et al. were also able to demonstrate and support the role of motivation as a mediator between the relationship of novelty need satisfaction and outcomes, which addresses the third criterion.{{vague}}{{example}}

===Interaction with the other three basic psychological needs=== 
Novelty-variety must be in [[w:synergy|synergy]] with the other three basic needs, rather than a need that only operates when the other basic needs are [[w:thwarted|thwarted]], frustrated or threatened (Fernandez - Espinola et al., 2020) . That is, novelty-variety must not be a [[w:substitute|substitute]] for one of the three basic needs. Studies show that the need for novelty-variety, along with the three existing basic psychological needs lead to motivation. González-Cutre et al. (2020) found that autonomous motivation was positively predicted by the satisfaction of the four needs, supporting the fourth criterion that the need for novelty works in synergy with the other basic psychological needs. Research provides that novelty-variety is a growth need rather than a [[w:deficit|deficit]] need.{{fact}}{{example}}

===Precursor of the increase of intrinsic motivation=== 
Ryan and Deci (2017) requires novelty-variety to be a precursor of growth, rather than an outcome of the natural, [[w:inherent|inherent]] growth process of intrinsic motivation. González-Cutre and Sicilia (2019) investigated the satisfaction of the three basic psychological needs and novelty in predicting adaptive outcomes directly and indirectly through intrinsic motivation. All 4 needs were positively [[w:correlated|correlated]] to intrinsic motivation, however, surprisingly, novelty-variety needs proved to be the best predictor of intrinsic motivation with respect to learning and understanding{{expand}}. This finding shows that novelty-variety is a precursor of intrinsic motivation. Therefore, the need for novelty-variety appears to have a similar function to the existing three needs and is a precursor of the inherent growth process of intrinsic motivation, satisfying the fifth criterion.{{example}}

===Universal operation===
Lastly, novelty-variety must operate universally regardless of [[w:sociocultural contexts|sociocultural contexts]]. Study conducted by Fernandez-Espinola et al. (2020) found that the effects of satisfying novelty-variety did not depend on the age of the participants or [[w:novelty-seeking|novelty-seeking]] preferences{{gr}}{{expand}}.

In further support, research to date has not found any gender differences in the effects of novelty–variety on various outcomes such as everyday happiness and life satisfaction (Sheldon et al. 2012). Research conducted in countries such as Spain, Germany and Canada strongly indicate that the effects of novelty–variety may surpass cultural boundaries (Bhageri, 2019).

Research to date supports novelty-variety in meeting Ryan and Deci’s (2017) sixth criterion.

{{RoundBoxTop|theme=2}}
'''Quiz yourself'''
&lt;quiz display=&quot;simple&quot;&gt;
{Who is considered as one of the earliest contributors of motivational psychology}
+ William James
- Martin Seligman
- Jay Shetty
- B. F. Skinner

{Free will was the first grand theory of motivational study. True or False}
+ True
- False
&lt;/quiz&gt;
{{RoundBoxBottom}}

==Evolutionary and physiological perspective==
Bhageri and Milyavskaya (2019) suggests that novelty-variety is important for human [[w:survival|survival]]. The innate desire to explore new and wide variety of resources has [[w:evolutionary|evolutionary]] benefits and those with a greater tendency towards novelty-variety are more likely to be associated with greater learning and intelligence. This positively impacts human survival as these individuals are more likely to find better sources of food and mates which, in turn, leads to better nutrition and chances of reproduction.

Research on [[w:dopamine|dopamine]] suggests that an encounter with a novel stimuli stimulates the [[w:hippocampus|hippocampus]] which releases dopamine. Dopamine is the reward that an individual will continuously seek. Neurobiological studies also suggest that novelty-variety seeking behaviour is largely associated with individuals’ [[w:neurotransmitter|neurotransmitter]] activities in the brain, specifically the release of dopamine (Li, 2020).

== Benefits to overall well-being ==
Novelty-variety has many implications for several positive outcomes in performance, life satisfaction, [[intrinsic motivation]], relationship satisfaction and quality.
Novelty-variety is linked with greater cognitive flexibility, creativity and to experiencing positive emotions in many areas of life.{{fact}}

Studies conducted by Sheldon &amp; Lyubomirsky (2012) suggests that novelty variety is positively associated with well-being.(17,18,19,20){{huh}} as variety, unexpectedness and surprise in everyday life promotes an increase in wellbeing.
Research conducted by Li et al. (2020) suggests that [[w:Exposure|exposure]] to novelty-variety can decrease one’s stress, anxiety and depression (22).{{fact}}

{{RoundBoxTop|theme=3}}
'''Food for thought:'''
* What factors affect an individuals{{gr}} [[w:self-motivation|self-motivation]]?{{vague}}
* Consider the [[w:psychosocial model|psychosocial model]] and it's benefits to aid motivation.{{vague}}
* How does one maintain an [[w:optimal|optimal]] level of motivation under difficult circumstances such as academics, gym workouts, daily work tasks, etc?{{vague}}{{ic|Focus reflection questions on novelty-variety}}
{{RoundBoxBottom}}

==Conclusion==
Novelty and variety factors have been studied extensively as properties of physical and [[w:mental wellbeing|mental wellbeing]] in wider contexts with novelty-variety being an emerging topic of research. As such, research in this area lacks a great depth of empirical and theoretical evidence and there are various [[w:limitations|limitations]] to the above findings. 

Some of the limitations relate to the need for further research to completely satisfy all six criteria for basic psychological needs. 

To entirely satisfy the third criterion for the basic psychological need within the self-determination theory, further research is required to determine whether satisfaction and frustration of novelty-variety mediates relation between social factors, motivation and outcomes. 

To entirely satisfy the sixth inclusion criterion within the self-determination theory, further research is required by doing similar research with a sample of non-western [[w:participants|participants]] as the current findings were determined by a group of western individuals. A sample of people of many different cultures is necessary to determine whether novelty-criteria is a universal psychological need and to promote [[w:generalisability|generalisability]] and transcend all cultural boundaries.{{fact}}

==See also==
* [[Motivation and emotion/Book/2022/Beneficence as a psychological need|Beneficence as a psychological need]] (Book chapter, 2022)
* [[Motivation and emotion/Book/2023/Health belief model|Heath belief model]] (Book chapter, 2023)
* [[wikipedia:Motivation|Motivation]] (Wikipedia)
* [[Motivation and emotion/Book/2023/Physiological needs|Physiological needs]] (Book chapter, 2023)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

==References==
{{Hanging indent|1=

Bagheri, L., &amp; Milyavskaya, M. (2019). Novelty–variety as a candidate basic psychological need: New evidence across three studies. ''Motivation and Emotion, 44''(1), 32–53. https://doi.org/10.1007/s11031-019-09807-4

Baumeister, R. F., &amp; Leary, M. R. (1995). The need to belong: Desire for interpersonal attachments as a fundamental human motivation. ''Psychological Bulletin, 117''(3), 497–529. https://doi.org/10.1037/0033-2909.117.3.497

Deci, E. L., &amp; Ryan, R. M. (2000). The “what” and “why” of goal pursuits: Human needs and the self-determination of behavior. ''Psychological Inquiry, 11''(4), 227–268. https://doi.org/10.1207/S15327965PLI1104_01

Fernández-Espínola, C., Almagro, B. J., Tamayo-Fajardo, J. A., &amp; Sáenz-López, P. (2020). Complementing the self-determination theory with the need for novelty: Motivation and intention to be physically active in physical education students. ''Frontiers in Psychology, 11'', Article 1535. https://doi.org/10.3389/fpsyg.2020.01535

González-Cutre, D., Romero-Elías, M., Jiménez-Loaisa, A., Beltrán-Carrillo, V. J., &amp; Hagger, M. S. (2019). Testing the need for novelty as a candidate need in basic psychological needs theory. ''Motivation and Emotion, 44''(2), 295-314. https://doi.org/10.1007/s11031-019-09812-7

González-Cutre, D., Sicilia, Á., Sierra, A. C., Ferriz, R., &amp; Hagger, M. S. (2016). Understanding the need for novelty from the perspective of self-determination theory. ''Personality and Individual Differences, 102'', 159–169. https://doi.org/10.1016/j.paid.2016.06.036

Li, W. W., Yu, H., Miller, D. J., Yang, F., &amp; Rouen, C. (2020). Novelty seeking and mental health in Chinese university students before, during, and after the COVID-19 pandemic lockdown: A longitudinal study. ''Frontiers in Psychology, 11'', Article 600739. https://doi.org/10.3389/fpsyg.2020.600739

Ryan, R. M., &amp; Deci, E. L. (2017). ''Self-determination theory: Basic psychological needs in motivation, development, and wellness''. Guilford Press. https://doi.org/10.1521/978.14625/28806

Sheldon, K. M., Elliot, A. J., Kim, Y., &amp; Kasser, T. (2001). What is satisfying about satisfying events? Testing 10 candidate psychological needs. ''Journal of Personality and Social Psychology, 80''(2), 325–339. https://doi.org/10.1037/0022-3514.80.2.325

Sheldon, K. M., &amp; Lyubomirsky, S. (2012). The challenge of staying happier. ''Personality and Social Psychology Bulletin, 38''(5), 670–680. https://doi.org/10.1177/0146167212436400

Siddique, N., Dhakan, P., Rano, I., &amp; Merrick, K. (2017). A review of the relationship between novelty, intrinsic motivation and reinforcement learning. ''Paladyn, Journal of Behavioral Robotics, 8''(1), 58–69. https://doi.org/10.1515/pjbr-2017-0004

Silvia, P. J. (2008). Interest—The curious emotion. ''Current Directions in Psychological Science, 17''(1), 57–60. https://doi.org/10.1111/j.1467-8721.2008.00548.x

Sylvester, B. D., Lubans, D. R., Eather, N., Standage, M., Wolf, S. A., McEwan, D., Ruissen, G. R., Kaulius, M., Crocker, P. R. E., &amp; Beauchamp, M. R. (2016). Effects of variety support on exercise-related well-being. ''Applied Psychology: Health and Well-Being, 8''(2), 213–231. https://doi.org/10.1111/aphw.12069

Vansteenkiste, M., Ryan, R. M., &amp; Soenens, B. (2020). Basic psychological need theory: Advancements, critical themes, and future directions. ''Motivation and Emotion, 44''(1), 1–31. https://doi.org/10.1007/s11031-019-09818-1

Wentworth, N., &amp; Witryol, S. L. (1983). Is variety the better part of novelty? ''Journal of Genetic Psychology, 142''(1), 3–15. https://doi.org/10.1080/00221325.1983.10533490

Zuckerman, M., &amp; Neeb, M. (1979). Sensation seeking and psychopathology. ''Psychiatry Research, 1''(3), 255–264. https://doi.org/10.1016/0165-1781(79)90007-6
}}

==External links==
* [https://selfdeterminationtheory.org/topics/application-basic-psychological-needs/ A basic psychological need]
* [https://www.mindtools.com/adosk97/how-self-motivated-are-you How motivated are you?]
*[https://selfdeterminationtheory.org/theory/ Self-determination theory]
*[https://www.verywellmind.com/what-is-self-determination-theory-2795387 Relationship between self-determination theory and motivation]
*[https://blog.idonethis.com/the-science-of-motivation-your-brain-on-dopamine/ Dopamine and motivation] 

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Needs]]
[[Category:Motivation and emotion/Book/Novelty]]</text>
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      <text bytes="21486" sha1="q358x7dtszsl23kl6r7994y5k5uci3r" xml:space="preserve">{{title|Psychological literacy:&lt;br&gt;What is psychological literacy, why does it matter, and how can it be fostered?}}
{{MECR3|1=https://www.youtube.com/watch?v=aXfQMx8tOt0&amp;ab_channel=LouiseG}}
__TOC__

==Overview==
{{RoundBoxTop|theme=2}}
[[File:A Happy Cartoon Businesswoman.svg|thumb|100px|'''Figure 1'''. Alex, the student who is considered psychologically literate.]] 

;Case study

Alex has a passion for studying the human mind and is about to graduate with an undergraduate psychology degree. Over the course of her studies, she has gained adequate skills and tools to be able to utilise her knowledge for psychological theories and apply it to everyday life. In her spare time, she has also been expanding her knowledge through extra study and work experience in various healthcare positions.

Through her knowledge and experience, she can now easily recognise the emotional state of herself and others and respond appropriately using psychological science. These skills have helped her reach her academic, professional and personal goals. She is very happy with her life. It is fair to say that Alex is a psychological literate citizen.
{{RoundBoxBottom}}

Psychological literacy is a modern psychological concept, which is defined as a person's ability to understand and regulate their own mind and behaviour, as well as recognising and responding to the emotional well being of others{{f}}. People experience a wide variety of emotions (as displayed in Figure 2), therefore becoming a Psychological Literate citizen can support the communication of not only one's self but also society.

[[File:Emotion collage.png|thumb|308x308px|Figure 2: Picture collage of a range of human emotions. Learning the skills to effectively understand and respond appropriately to emotions, develops a person’s psychological literacy. ]]

Equipping people with the skills and tools to understand and regulate their own feelings and behaviour can be beneficial for not only the individual but also for society. Psychological literacy also has many immediate, short term and long term benefits and if the approach is introduced to people early in life, can offer better academic, social and mental health outcomes. 

However this psychological concept is still in the early stages of development, with questions regarding the reliability and validity of this concept needing to be investigated. Further research regarding the social benefit of if and how it can be implemented in today's society is required.

{{robelbox|theme=9|title=Focus questions:|icon=Nuvola_apps_kwrite.png|iconwidth=48px}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
* What is psychological literacy?
* Why is psychological literacy important?
* How can psychological literacy be fostered?
&lt;/div&gt;
{{Robelbox/close}}

==What is Psychological Literacy?==
Psychological literacy is a person's ability to utilise their knowledge of psychological theory in practice, primarily in everyday life. This can be measured through the assessment of specific key concepts that a person displays such as self-management, resilience and use of positive psychology.

=== Definition ===
Psychological literacy as defined by Cranney et al. (2012, piii) is “the general capacity to adaptively and intentionally apply psychology to meet personal, professional, and societal needs&quot;. In simpler terms, it is the capacity that an individual person has to self-manage their emotions and to recognise the emotional state of others, in order to produce positive social outcomes. 

=== Brief History of Psychological Literacy ===
The History of psychological literacy is quite brief, as it is a relatively new concept. The American psychologist Alan Boneau, only coined the term 'psychological literacy' in 1990 (Boneau, 1990). During his research, Boneau was looking to define key concepts in multiple psychological subfields that would be most useful for those in the psychological society, especially students. This is why he then developed the term ‘psychological literacy’, which has individual subsections that contain multiple key concepts, to be measured in order to assess the level of PL a person has. Boneau collected research through enquiries and questionnaires from other researchers in the field, to adequately rate and organise the key concepts in order of importance and relevance (Boneau, 1990). Although Boneau’s research was extensive and produced the basis of PL as a concept, he emphasised the importance of further development in this area and welcomed further suggestions. 

Despite Boneau’s research, this concept was sparsely used in mainstream psychology. This was until 2010 when more interpretations of Boneau’s research was expanded upon and used. During this period the term was more specifically used to describe graduates of psychological education (Harris et al., 2021). These are often referred to as graduate attributes, such as knowledge, skills and attitudes.

However, during the 2020s, psychological literacy has been generally applied to the global population. Further research has been undertaken on the effectiveness of teaching these concepts to young people and children and how to incorporate this learning into age-appropriate language through the school curriculum (Cranney et al., 2022b). Since the increase of interest in PL during the 2010's{{g}}, more studies have been conducted to test the validity and reliability of this psychological concept.

=== Concepts of Psychological Literacy ===
{{expand}}

Psychological literacy consists of these key concepts/theories
{| class=&quot;wikitable&quot;
|+
!Key concepts 
!Definitions
|-
|[[Psychological resilience|Resilience]] 
|Through learned optimism, an individual can build resilience.
|-
|[https://www.betterup.com/blog/what-is-self-management-and-how-can-you-improve-it Self-management] 
|The process where an individual learns healthy ways to manage and self-regulate their emotional well-being.
|-
|[[Positive psychology]] 
|Is an approach to mental and emotional well being through the balance of positive and negative experiences. Effective coping strategies and emotional skill development are used to create long term positive psychological health.
|-
|[https://www.verywellmind.com/what-is-adaptation-2794815 Adaptive cognition theory]
|Is defined as global ways of thinking and consequently behaving that are beneficial to one's (and others) survival and well-being.
|}

== Why is Psychological Literacy Important? ==
Having psychological literacy skills, whether basic or advanced, can be beneficial for not only the individual but other aspects of society. For example, parents and carers of children may find that having PL, benefits them when interacting and parenting children. Also, if children are taught from early childhood how to manage their emotions, as they grow up they are less likely to experience antisocial behaviours which then benefits society.{{f}}

=== Importance for Individuals ===
Early intervention of psychological literacy during childhood equips the child with appropriate tools to learn how to self-regulate and manage their emotional well-being. These tools help children learn to self soothe which in turn builds resilience and empathy which is essential for adults. Therefore, as the child enters adolescence and eventually adulthood, having the foundational skills for resilience and self-management already gives them an advantage to achieve their personal, social and professional goals. Studies have also shown that people with high levels of emotional literacy experience lower rates of mental illness (Bezzina &amp; Camilleri, 2021){{expand}}. 

=== Importance for Parents and Carers of Children ===
[[File:Empathy 1.jpg|thumb|231x231px|Figure 3 : An image of two woman{{sp}} showing empathy and love towards a young child. This helps the child's development of Empathy.]]

Raising and parenting children can, at times, be a challenging and overwhelming experience. Having a good understanding of how and why children may behave and express themselves benefits not only the child but also the guardian (Black &amp; Trude 2019). A psychologically literate would have a basic understanding of developmental psychology, therefore would have appropriate knowledge to support the child/children through the different stages of their childhood.   

This is essential when supporting children with the development of [[Empathy Models|empathy]], which begins in early childhood. Empathy is a crucial part of human emotion, helping us connect with others and function effectively in society. It is a vital part of building and establishing relationships and avoiding conflict (Schonert-Reichl, 2013). Therefore, supporting children in the development of empathy in early childhood, as shown in Figure 3, supports them in learning how to have healthy relationships within families and in furthermore in their future relationships{{g}}. 

=== Importance for Society ===
The importance of psychological literacy for society is that if a person has high PL they are more likely to be a productive member of society. The more productive members of a society, the stronger the economy and labour market which reduces the amount of people living in lower socio-economic lifestyles and reduces antisocial behaviour (Brindle et al., 2019). Higher levels of psychological literacy also increases academic/career outcomes which in turns reduces poverty.

== The Benefits of Psychological Literacy ==
Psychological literacy offers benefits that are short-term and long-term, as the skills learned can be utilised from early childhood into adulthood. 

=== Short Term Benefits ===
The short-term benefits for children, if introduced in early childhood, is that PL can reduce the occurrence/frequency of tantrums and emotional outbursts. As the children are already developing the skills for PL, they can be more independent and focus on building healthy relationships and express themselves in an effective and appropriate manner (Lincoln et al., 2017). This also benefits parents and carers in the short-term.

When a person experiences emotional distress during adolescence and adulthood, they would also already have the tools and skills to manage their emotions which would in turn reduce the negative health outcomes. This would place less stress on the healthcare system and offer better performance in social settings, workplaces/schools and relationships (Cranney et al., 2022b). 

=== Long Term Benefits ===
Individuals who can self-manage their emotional well-being, have a better chance of completing their education, forming healthy relationships and becoming productive members of society (Olympia et al., 1994). Productive members of society are also less likely to engage in anti-social behaviour. Therefore, learning these skills in childhood are beneficial in the long-term outcomes{{g}}.

Increasing the occurrence of PL in childhood would then reduce anti-social behaviour and criminal activity, as the individuals who engage in these activities often aren't equipped with the skills to self-regulate their emotional/mental well-being (Gaik et al., 2010). Through the implementation of proficient psychological literacy, the local economy with{{sp}} benefit through the increase in productivity and the decrease in poverty.

A study conducted by James (2011), {{gr}} found that law students experienced a high rate of mental illnesses related to the profession. These include high rates of depression, anxiety and work burnout. However, when offered psychological literacy education including an early intervention of mindfulness and self-management training, the students experienced lower rates of mental disorders related to their profession during their studies but most importantly in the long-term{{expand}}. 

== How can Psychological Literacy be Fostered? ==
One approach to fostering psychological literacy is to incorporate this concept into the health studies of the school curriculum, from early childhood through to adolescence (Hulme &amp; Cranney, 2021). Teaching children about positive psychology and mindfulness will support them with self-managing their emotional well-being (Hulme &amp; Taylor, 2015). Intervention of psychological literacy during early childhood education will be beneficial, as this is a key developmental period in a child's life where they are learning how to navigate social interactions.

A study conducted by Arslan et al. (2022) measured the effectiveness of storytelling, using positive psychology literature, on the well-being of high school students. They found that the students who took part in mindfulness storytelling experienced an improvement in  mindfulness, optimism, happiness, and positive emotions, and a reduction in depression, anxiety, pessimism, and other negative emotions over a 5-week period (Arslan et al., 2022). Therefore, incorporating PL training into the curriculum through storytelling would be an effective approach.  

== Criticisms of Psychological Literacy ==
We have discussed why psychological literacy is important as well as the short- and long-term benefits associated with adequate PL however it is still a relatively new concept. The validity and reliability of the concept need to be critically analysed. There is also no framework for the appropriate/adequate delivery of psychological literacy for either children or adults. Most research has focused on the benefits for persons in the psychology field rather than the general population.

It is not realistic to expect that people with PL can ‘fix’ issues/problems with their knowledge. It can also be harmful if overused in everyday life and in all human interactions{{f}}. Some people may find it challenging and exhausting to translate psychological theory into all real-world situations (Pownall, 2017). This can create unrealistic expectations for those who are educated in psychology to constantly evaluate and provide solutions to psychological issues in the real-world.

{{RoundBoxTop|theme=3}}
[[File:Noun emotion 1325508.svg|thumb|Figure 4 : Alex is feeling sad and overwhelmed ]]

;Case study
Although Alex is considered a psychological literate citizen, it is becoming increasingly exhausting for her to always be expected to ‘fix’ problems using her skills and knowledge. Alex is a supportive person who likes to help her friends and family, however those around Alex are now always expecting her to provide theory based advise. Her psychological literacy has now become a burden and is impacting how Alex is interacting and spending time with people. 
{{RoundBoxBottom}}

==Conclusion==
Psychological literacy has validity for adults to be able to adequately self-manage their emotion well-being. Teaching PL during early childhood and through to adulthood has many short- and long-term benefits including a healthy development of Empathy for children and young people which in turn offers better mental health and well-being outcomes. This can be incorporated into the school curriculum and effectively delivered through fictional storytelling. However, this concept is still relatively new and further research is needed to provide conclusive evidence of the role that psychological literacy plays in today’s society. Further research is also needed to produce {{missing}} framework for the appropriate/adequate delivery of psychological literacy for children and adults.

==See also==
* [[Emotional intelligence]] (Wikiveristy{{sp}}) 
* [[Motivation and emotion/Book/2019/Emotional intelligence and anti-social behaviour|Emotional intelligence and anti-social behaviour]] (Book chapter, 2019)
* [[Motivation and emotion/Book/2013/Emotional self-regulation|Emotional regulation]] (Book chapter, 2013) 
* [[Empathy Models/Empathy Model|Empathy]]  (Wikiversity) 
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[wikipedia:Psychological_literacy|Psychological literacy]] (Wikipedia) 
* [[w:Self determination theory|Self determination theory]] (Wikipedia)

==References==
{{Hanging indent|1=

Arslan, G., Yıldırım, M., Zangeneh, M., &amp; Ak, İ. (2022). Benefits of positive psychology-based story reading on adolescent mental health and well-being. Child indicators research, 15(3), 781-793. https://doi.org/10.1007/s12187-021-09891-4

Bezzina, A., &amp; Camilleri, S. (2021). ‘Happy Children’A project that has the aim of developing emotional literacy and conflict resolution skills. A Maltese Case Study. Pastoral care in education, 39(1), 48-66.  &lt;nowiki&gt;https://doi.org/&lt;/nowiki&gt;[https://www.tandfonline.com/doi/full/10.1080/02643944.2020.1774633 10.1080/02643944.2020.1774633] 

Black, M. M., &amp; Trude, A. C. (2019). Conceptualizations of child development benefit from inclusion of the nurturing care framework. The ''Journal of nutrition'', 149(8), 1307–1308. https://doi.org/10.1093/jn/nxz114 

Boneau, C. A. (1990). Psychological literacy: A first approximation. ''American Psychologist'', ''45''(7), 891. https://doi.org/10.1037/0003-066X.45.7.891 

Brindle, K. A., Bowles, T. V., &amp; Freeman, E. (2019). Gender, education and engagement in antisocial and risk-taking behaviours and emotional dysregulation. Issues in Educational Research, 29(3), 633–648. https://www.iier.org.au/iier29/brindle.pdf

Cranney, J., Botwood, L., &amp; Morris, S. (2012a). National standards for psychological literacy and global citizenship: Outcomes of undergraduate psychology education. ''Australia: The University of New South Wales. Dostupné z:''  https://groups.psychology.org.au/assets/files/cranney_ntf_final_report_231112_final_pdf.pdf 

Cranney, J., Dunn, D. S., Hulme, J. A., Nolan, S. A., Morris, S., &amp; Norris, K. (2022b). Psychological literacy and undergraduate psychology education: An international provocation. In Frontiers in Education (Vol. 7, p. 790600). Frontiers Media SA. https://doi.org/10.3389/feduc.2022.790600 

Gaik, L. P., Abdullah, M. C., Elias, H., &amp; Uli, J. (2010). Development of antisocial behaviour. Procedia-Social and Behavioral Sciences, 7, 383–388. https://doi.org/10.1016/j.sbspro.2010.10.052 

Harris, R., Pownall, M., Thompson, C., Newell, S. J., &amp; Blundell-Birtill, P. (2021). Students' Understanding of Psychological Literacy in the UK Undergraduate Curriculum. Psychology Teaching Review, 27(1), 56–68. https://files.eric.ed.gov/fulltext/EJ1304625.pdf 

Hulme, J. A., &amp; Cranney, J. (2021). Psychological literacy and learning for life. In International handbook of psychology learning and teaching (pp. 1-29). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-030-26248-8_42-2

James, C. (2011). Law student wellbeing: Benefits of promoting psychological literacy and self-awareness using mindfulness, strengths theory and emotional intelligence. Legal Education Review, 21(1/2), 217–233. https://search.informit.org/doi/pdf/10.3316/informit.868049526963988 

Lincoln, C. R., Russell, B. S., Donohue, E. B., &amp; Racine, L. E. (2017). Mother-child interactions and preschoolers’ emotion regulation outcomes: Nurturing autonomous emotion regulation. Journal of Child and Family Studies, 26, 559–573. https://doi.org/10.1007/s10826-016-0561-z

Olympia, D. E., Sheridan, S. M., Jenson, W. R., &amp; Andrews, D. (1994). Using student‐managed interventions to increase homework completion and accuracy. Journal of Applied Behavior Analysis, 27(1), 85–99. https://doi.org/10.1901/jaba.1994.27-85  

Pownall. M.. (2017, June 27). ''Overrated: Psychological Literacy.'' The British Psychological Society. https://www.bps.org.uk/psychologist/overrated-psychological-literacy  

Schonert-Reichl, K. (2013). Promoting empathy in school-aged children: Current state of the field and implications for research and practice. In School rampage shootings and other youth disturbances (pp. 159–203). https://www.researchgate.net/profile/Kimberly-Schonert-Reichl/publication/299566650_Promoting_empathy_in_school-aged_children_Current_approaches_and_implications_for_practice/links/56ffc38d08aee995dde81820/Promoting-empathy-in-school-aged-children-Current-approaches-and-implications-for-practice.pdf 

Taylor, J., &amp; Hulme, J. (2015). Introducing a compendium of psychological literacy case studies: Reflections on psychological literacy in practice. ''Psychology Teaching Review'', ''21''(2), 25-34. https://files.eric.ed.gov/fulltext/EJ1146628.pdf 
}}

==External links==
* [https://www.discovermagazine.com/mind/how-reading-fiction-increases-empathy-and-encourages-understanding How reading fiction encourages empathy] (Discover Magazine) 
* [https://www.apa.org/topics/mindfulness Mindfulness psychology] (American Psychological Association) 
* [https://www.blackdoginstitute.org.au/wp-content/uploads/2022/06/Positive-psychology-fact-sheet.pdf Positive psychology] (Blackdog Institute) 
* [https://positivepsychologytraining.co.uk/ Positive psychology training uk] (Positive psychology Training) 
* [https://www.bps.org.uk/psychologist/psychological-literacy-classroom-real-world Psychological literacy – from classroom to real world] (British Psychological Society)
* [https://www.apa.org/topics/resilience/guide-parents-teachers Resilience guide for parents and teachers] (American Psychological Association)
* [https://www.berkeleywellbeing.com/self-management.html Self-management] (Berkley Well-being Institute) 

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{{title|Functional motives theory:&lt;br&gt;How does this theory link to environmental activism?}}
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==Overview==
Environmental activists are passionate individuals, who are strongly committed to positively influencing a change for the environment. Although environmental activists are highly committed to their work, the pace of the change they want to achieve can become painful and frustrating. There are many uncertainties that arise when following a path of environmental activism, such as 'will this benefit the environment in 20 years time?' or 'how do we know this will implement a positive change?'. In knowing this, we ask the question, how do environmental activists stay motivated and committed to their work with the uncertainty and frustration that what they are doing may not influence beneficial change?{{RoundBoxTop|theme=5}}[[Scenario]]
[[File:A picture is worth a thousand words.jpg|right|thumb|170px|'''Figure 1'''. Explore the topic, then brainstorm a structure.]]
An environmental activist is adamant on reducing environmental pollution such as pollution in the air and water. The activist has suggested peaceful protests as well as flyers around the city she lives in to engage more individuals in the cause. She has decided to leave a link to a website she created in hopes others will sign up and join her to help make a change. After one week, she noticed there had only been three people sign up, which is not enough to start a protest. The activist feels hopeless and uncertain that her initiatives are not enough to contribute to a sustainable environment.
So how can we assure the activist remains motivated?
{{RoundBoxBottom}}

This template provides tips for the [[Motivation and emotion/Assessment/Topic|topic development]] exercise. Gradually remove these suggestions as the chapter develops. It is OK to retain some of this template content for the topic development exercise. Also consult the [[Motivation and emotion/Assessment/Chapter|book chapter guidelines]].

The Overview is typically consists of one to four paragraphs inbetween the scenario and focus questions. Suggested word count aim for the Overview: 180 to 330 words.
{{tip|Suggestions for this section:
* Engage the reader with a scenario, example, or case study, and an accompanying image
* Explain the problem and why it is important
* Outline how psychological science can help
* Present focus questions
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'''Focus questions:'''
Break the problem (i.e., the sub-title) down into three to five focus questions. Focus questions can also be used as top-level headings.
* What is the first focus question?
* What is the second focus question? 
* What is the third focus question?

Ask [[w:Open-ended question|open-ended]] focus questions. For example:
* Is there a relationship between motivation and success? (closed-ended) {{sad}}
* What is the relationship between motivation and success? (open-ended) {{smile}}
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==Headings==
* Aim for three to six main headings inbetween the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]
* Sub-headings can also be used, but
** avoid having sections with only one sub-heading
** provide an introductory paragraph before breaking into sub-sections

==Key points==
* Provide at least three bullet-points per headingʔ and sub-heading, including for the Overview and Conclusion
* Include key citations

==Figures==
[[File:Thought bubble.svg|right|140px|thumb|'''Figure 2'''. Example of an image with a descriptive caption.]]

* Use figures to illustrate concepts, add interest, and to serve as examples
* Figures can show photos, diagrams, graphs, video, audio, etcetera
* Embed figures throughout the chapter, including the Overview section
* Figures should be captioned (using '''Figure #.''' and a caption). Use captions to explain the relevance of the image to the text/
* [[commons:|Wikimedia Commons]] provides a library of embeddable images
* Images can also be uploaded to [[commons:|Wikimedia Commons]] if they are openly licensed
* Refer to each figure at least once in the main text (e.g., see Figure 2)

==Learning features==
Interactive learning features help to bring online book chapters to life and can be embedded throughout the chapter.

{{anchor|Scenarios}}
;Scenarios
* Scenarios or case studies describe applied/real-world examples of concepts in action
* Case studies can be real or fictional
* A case study could be split into multiple boxes throughout a chapter (e.g., to illustrate different theories or stages)
* It is often helpful to present case studies using [[#Feature boxes|feature boxes]].

{{anchor|Feature box}}
;Feature boxes
* Important content can be highlighted in a [[Motivation and emotion/Wikiversity/Feature box|feature box]]. But don't overuse feature boxes, otherwise they lose their effect.
* Consider using feature boxes for:
** [[#Scenarios|Scenarios]], case studies, or examples
** Focus questions
** Tips
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;Links
* When key words are introduced, use [[Help:Links|interwiki links]] to:
** Wikipedia (e.g., [[w:Sigmund Freud|Sigmund Freud]] wrote about (e.g., [[w:Dreams|dreams]]) or
** Related book chapters (e.g., [[Motivation and emotion/Book/2020/Writer's block|writer's block]])

;Tables
* Use to organise and summarise information
* As with [[#Figures|figures]], tables should be captioned
* Refer to each table at least once in the main text (e.g., see Table 1)
* [[Motivation and emotion/Wikiversity/Tables|Example 3 x 3 tables]] which could be adapted

'''Table 1.''' Descriptive Caption Which Explains The Table and its Relevant to the Text - Johari Window Model

{| class=&quot;wikitable&quot; style=&quot;margin: auto;
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! !! Known to self !! Not known to self
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| '''Not known to others''' || Hidden area || Unknown
|}

;Quizzes
* Using one or two review questions per major section is usually better than a long quiz at the end
* Quiz ''conceptual'' understanding, rather than trivia
* Don't make quizzes too hard
* Different types of quiz questions are possible; see [[Help:Quiz|Quiz]]

Example simple quiz questions. Choose your answers and click &quot;Submit&quot;:
&lt;quiz display=simple&gt;
{Quizzes are an interactive learning feature:
|type=&quot;()&quot;}
+ True
- False

{Long quizzes are a good idea:
|type=&quot;()&quot;}
- True
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==Conclusion==
* The Conclusion is arguably the most important section
* Suggested word count: 150 to 330 words
* It should be possible for someone to only read the [[#Overview|Overview]] and the Conclusion and still get a pretty good idea of the problem and what is known based on psychological science
{{tip|Suggestions for this section:
* What is the answer to the sub-title question based on psychological theory and research?
* What are the answers to the focus questions?
* What are the practical, take-home messages? (Even for the topic development, have a go at the likely take-home message)
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==See also==
Provide [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to the most relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. Use these formats:
* [[Motivation and emotion/Book/2021/Light triad|Light triad]] (Book chapter, 2021)
* [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki]] (Wikiversity)
* [[w:Self determination theory|Self determination theory]] (Wikipedia)
{{tip|Suggestions for this section:
* Present in alphabetical order
* Use [[w:Letter case#Sentence casing|sentence casing]]
* Include the source in parentheses
}}

==References==
List cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]].

APA style example:
{{Hanging indent|1=

Rosenberg, B. D., &amp; Siegel, J. T. (2018). A 50-year review of psychological reactance theory: Do not read this article. ''Motivation Science'', ''4''(4), 281–300. https://doi.org/10.1037/mot0000091
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{{tip|Suggestions for this section:
* Important aspects of APA style for references include:
** Wrap the set of references in the [[Template:Hanging indent|hanging indent template]]. Use &quot;Edit source&quot;: &lt;nowiki&gt;{{Hanging indent|1= the full list of references}}&lt;/nowiki&gt;
** Author surname, followed by a comma, then the author initials separated by full stops and spaces
** Year of publication in parentheses
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** Provide the full doi as a URL and working hyperlink
* The most common mistakes include:
** Incorrect capitalisation
** Incorrect italicisation
** Citing sources that weren't read or consulted
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==External links==
Provide [[Help:Contents/Links#External_links|external links]] to highly relevant resources such as presentations, news articles, and professional sites. Use [[w:Letter case#Sentence casing|sentence casing]]. For example:
* [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne)
* [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com)
{{tip|Suggestions for this section:
* Only select links to major external resources about the topic
* Present in alphabetical order
* Include the source in parentheses after the link
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==See also==
* [[Motivation and emotion/Book/2024/Power motivation and leadership|Power motivation and leadership]] (Book chapter, 2024)

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==Overview ==
{{Robelbox|width=30|theme={{{theme|3}}}|title=Case study: Sarah's story: Part 1}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;

Sarah, a 29-year-old woman, survived a car accident (see Figure 1) unscathed caused by a drunk driver{{g}}. However{{g}} she blames herself for the severe injuries that her best friend endured. Despite clear evidence confirming that she was not at fault, Sarah can't shake the constant guilt, fixating on the moment and her own actions, believing she could have prevented the crash.

&lt;/div&gt;
{{Robelbox/close|theme=6}}

[[w:Self-blame|Self-blame]] poses a significant psychological barrier to emotional recovery from trauma. It occurs when individuals wrongly attribute an incident to their own behaviour or perceived personal weaknesses. Self-blame can be categorised into two forms adaptive (behavioural) self-blame, and maladaptive (characterological) self-blame (Janoff-Bulman, p. 979). While behavioural self-blame can represent an attempt to regain control and involves attributing the event to specific actions or decisions that are perceived as controllable (e.g., “I shouldn’t have gone there alone”), characterological self-blame targets stable, internal traits as a symptom of personal defectiveness such as the type of person they are (e.g., “I’m the kind of person bad things happen to”), this is the most damaging type of self-blame. 

When [[trauma]] disrupts emotional regulation, self-image, self-blame can exacerbate feelings of [[w:Guilt|guilt]], fear, [[w:Anxiety|anxiety]], anger, and helplessness. Survivors who suffer such trauma may resist support and feel undeserving of care, find it difficult to understand what they have been through, and not ask for help. 

Fortunately, therapies such as [[w:Cognitive processing therapy|cognitive processing therapy]] (CPT), self-compassion interventions, and narrative approaches are powerful ways to challenge these distorted beliefs and re-establish a more compassionate and coherent self-narrative. By dismantling the self-perpetuating cycles of blame and emotional suppression, therapists can empower individuals to engage more fully in their recovery journey, enhancing emotional healing and long-term psychological well-being.

{{RoundBoxTop|theme=6}}
'''Focus questions'''

* How does self-blame influence emotional and psychological recovery after traumatic experiences?
* What cognitive and emotional processes underlie self-blame responses? 
* How can therapeutic interventions address maladaptive and adaptive self-blame to support recovery?
* How does self-blame affect and present itself within different demographics?
{{RoundBoxBottom}}

==Understanding self-blame in the context of trauma==
Self-blame is a cognitive process in which individuals attribute the cause of a negative or traumatic event to themselves, often as a way to make sense of overwhelming experiences (Jacobsen &amp; Petersen, 2023). These {{what}} differ not only in their attributions but also in their psychological consequences and the therapeutic strategies used to address them (see Table 1). Recognising the distinction between the types of self-blame such as characterological and behavioural is essential for understanding why some trauma survivors struggle to recover emotionally. It is also critical in tailoring interventions such as Cognitive Processing Therapy (CPT), which specifically targets maladaptive self-blame patterns (Resick et al., 2002).

{| class=&quot;wikitable&quot;
|+'''Table 1.''' Types of Self-Blame, Their Psychological Effects, and Therapeutic Interventions
!Types of Self Blame
!Psychological/Emotional Effects
!Therapeutic Approaches
|-
|Behavioural Self-Blame (attributing trauma to specific controllable actions)       
|May foster a sense of control; can sometimes be adaptive if it motivates constructive coping.
|Psychoeducation, adaptive attribution reframing.
|-
|Characterological Self-Blame (attributing trauma to stable internal traits)
|Strongly linked to shame, depression, PTSD, withdrawal, and chronic rumination.
|Cognitive processing therapy (CPT), compassion-focused therapy (CFT), narrative therapy, mindfulness interventions.
|}

[[File:Sadness at the beach.jpg|174x174px|thumb|'''Figure 2'''. Emotional reaction to a traumatic experience.]]

In the aftermath of trauma such as [[w:Sexual assault|sexual assault]], [[w:Child abuse|childhood abuse]], [[w:Natural disaster|natural disasters]], and [[w:Domestic violence|domestic violence]], self-blame frequently emerges as a [[w:Coping (psychology)|coping]] mechanism (Berman et al., 2018). Reports show that up to 74% of sexual assault survivors report some form of self-blame, with characterological self-blame being especially common among those who experienced abuse in childhood (Janoff-Bulman, 1978). Self-blame can provide survivors with a distorted sense of control, helping them believe that if they caused the trauma, they can prevent future harm. However, this coping strategy often backfires, deepening emotional distress and impeding recovery. See Figure 2 for a visual representation of emotional distress following a traumatic experience. In contexts like assault or loss, survivors may internalize blame to preserve relationships or avoid confronting the randomness and cruelty of the event, especially when the perpetrator is someone they trusted. (Overstreet &amp; Quinn, 2013)This internalization can lead to chronic [[w:Shame|shame]], depression, and difficulty seeking support.

Demographic factors play a significant role; for example, women and younger individuals are more likely to engage in self-blame, possibly due to societal conditioning around guilt and responsibility (Brown, 2013). Trauma severity also influences self-blame; those who endure prolonged or repeated trauma often internalize blame more deeply (Melville et al., 2014). Higher levels of depression and lower emotional well-being are reported by patients with a higher characterological self-blame rate than a behavioural self-blame rate (Pham et al., 2021). A 2020 study on survivors of childhood neglect who used self-blame as a coping strategy found that they were more likely {{ic|than?}} to develop internalising behaviours such as anxiety and depression, with an effect size of .28, indicating a moderate impact (Tanzer et al., 2020). These findings underscore the psychological toll of self-blame and highlight the need for trauma-informed interventions that address its roots and manifestations across diverse populations.

==Cognitive and emotional mechanisms underlying self-blame==
{{Robelbox|width=30|theme={{{theme|3}}}|title=Case study: Sarah's story: Part 2}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;

[[File:Traffic at Viaduto Tutóia.jpg|Traffic_at_Viaduto_Tutóia|The Daunting lines of Traffic - Viaduto Tutóia|right||174x174px|thumb|'''Figure 1'''. The Daunting lines of Traffic.]]

Months later, Sarah is still experiencing persistent feelings of guilt, depression, and panic attacks towards driving on the road. Before the accident, Sarah was a confident social person however, following the incident, she avoids driving and social contact entirely, believing she's undeserving of support. Her emotional withdrawal and negative self-appraisals reflect a deepening cycle of self blame.

&lt;/div&gt;
{{Robelbox/close|theme=6}}

=== Attribution theory and self-blame ===
[[w:Attribution theory|Attribution theory]] explores how individuals interpret the causes of events, particularly in the aftermath of trauma (Malle, 2011). Survivors often fluctuate between internal attributions, blaming themselves for the trauma, and external attributions, which place responsibility on outside forces such as perpetrators or circumstances. Internal attributions, especially when tied to characterological traits (e.g., “I’m weak,” “I deserved it”), are strongly linked to depression and anxiety, as they reinforce feelings of helplessness and low self-worth (Peterson et al., 1981).

In contrast, external attributions may offer psychological relief but can also provoke anger or fear. Research shows that [[w:Attribution (psychology)|attribution (psychology)]] styles significantly influence mental health outcomes: individuals who consistently attribute trauma to internal, stable causes are more likely to experience PTSD symptoms{{f}}. This is known as [[w:Attribution bias|attribution bias]], where the mind habitually interprets events through a self-critical lens (Tian et al., 2020). Many studies show that trauma survivors with high internal attribution tendencies have a greater risk of developing PTSD compared to those with more balanced attribution styles (Massad &amp; Hulsey, 2006). These cognitive patterns not only shape how survivors understand their trauma but also determine the emotional and behavioural strategies they adopt in recovery.

=== Emotional responses driving self-blame ===
Self-blame is deeply intertwined with emotional responses such as shame and guilt, which play distinct roles in trauma recovery (Jacobsen &amp; Petersen, 2023). Guilt arises from perceived wrongdoing and can motivate reparative action, while shame targets the self, leading to feelings of worthlessness and isolation. Shame is particularly corrosive, often driving emotional avoidance and [[w:Intrusive thought|intrusive thoughts]] that reinforce the trauma narrative.

Survivors may engage in [[w:Dissociation|dissociation]], a psychological detachment from reality as a way to escape overwhelming emotions, yet this can paradoxically intensify self-blame by disconnecting them from external validation or support (Drescher, 2022). Rumination, the repetitive and passive focus on distressing thoughts, further entrenches blame cycles (Moulds et al., 2020). Studies show that individuals who ruminate excessively are more likely to maintain negative self-schemas, deeply held beliefs about being flawed or unworthy, which perpetuate self-blame and hinder emotional healing. For example, a 2024 study found that high rumination levels strengthened the indirect effect of childhood trauma on depression, by altering how sense of control mediates the relationship (You, et al., 2024). These emotional and cognitive mechanisms form a feedback loop: self-blame fuels shame and rumination, which in turn reinforce negative beliefs and [[w:Emotional dysregulation|emotional dysregulation]]. Breaking this cycle requires therapeutic interventions that target both attribution biases and maladaptive emotional responses, fostering self-compassion and [[w:Cognitive restructuring|cognitive restructuring]].

==Impact of self-blame on emotional recovery==
{{ic|Include an introductory paragraph before branching into sub-sections}}

=== Psychological consequences ===
Self-blame following trauma is associated with a range of adverse psychological outcomes, including heightened risks of depression, anxiety, and PTSD. Survivors who internalize blame often experience a profound disruption in their self-concept and identity, viewing themselves as fundamentally flawed or responsible for their suffering (Hyland et al., 2023). This distorted self-perception can lead to chronic emotional dysregulation, where individuals struggle to manage intense feelings such as shame, guilt, and fear (Gratz &amp; Roemer, 2004). Over time, these emotional burdens contribute to lower treatment-seeking behaviour, as survivors may feel undeserving of help or fear judgment from others (Wang, 2023). According to Psychology Today, self-blame acts as a form of emotional self-abuse, reinforcing perceived inadequacies and paralyzing individuals before they can begin healing (Formica, 2013). This internalised shame often results in interpersonal withdrawal, where survivors isolate themselves to avoid vulnerability or perceived rejection (Gilbert, 2000). The cumulative effect is a cycle of psychological distress that not only impairs daily functioning but also entrenches negative beliefs about the self, making recovery more difficult (Michelle, 2020). Studies show that trauma survivors who engage in characterological self-blame, blaming their inherent traits rather than specific actions, are more likely to develop persistent depressive symptoms and exhibit lower resilience in the face of future stressors. 

=== Disruption to recovery processes ===
Self-blame significantly disrupts the recovery process by obstructing the survivor’s ability to integrate the trauma into their life narrative and derive meaning from the experience. Instead of fostering growth or acceptance, self-blame reinforces avoidance behaviours and [[w:Hypervigilance|hypervigilance]], keeping the nervous system in a state of chronic alert (Ullman &amp; Filipas, 2001). This impairs emotional processing and prolongs symptoms such as intrusive thoughts and [[w:Flashback (psychology)|flashbacks]], which are hallmarks of PTSD (Foa &amp; Rothbaum, 1998). Survivors may become trapped in a loop of rumination and emotional suppression, unable to confront or reframe the trauma constructively. Moreover, self-blame weakens interpersonal relationships and [[w:Social support|social support]] networks, as individuals may feel unworthy of connection or fear being misunderstood (Ullman &amp; Peter-Hagene, 2014). This isolation further compounds emotional distress and delays healing.

In clinical settings, self-blame can block the formation of a therapeutic alliance, a crucial component of effective trauma therapy. Survivors may resist vulnerability or distrust the therapist, believing their suffering is self-inflicted or deserved. Survivors of childhood trauma often carry toxic shame and chronic self-criticism into adulthood, which hinders emotional regulation and makes it difficult to seek or accept help (Duarte, 2017). Addressing self-blame is therefore essential not only for symptom relief but also for restoring a sense of agency, connection, and hope in the recovery journey.

{{Robelbox|theme=5|width=100%|title=Test your Knowledge !|iconwidth=48px|icon=Nuvola_apps_korganizer.svg}}&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
&lt;quiz display=simple&gt;
{ Fill in the blanks: Behavioral self-blame attributes negative events to________, while characterological self-blame targets_______. }

|type=&quot;()&quot;}

+ Specific actions or decisions, Stable internal traits 
- Random events, Personal relationships
- Uncontrollable circumstances, External factors
- External factors, Internal traits

{What is a key psychological consequence of characterological self-blame? }

|type=&quot;()&quot;}

- Increased resilience to future stressors
- Enhanced Interpersonal relationships
+ Lower treatment-seeking behaviour 
- Improved emotional regulation

{What is a common coping mechanism that survivors use to escape overwhelming emotions, which can paradoxically intensify self-blame?  }

|type=&quot;()&quot;}

- Rumination
+ Dissociation
- Avoidance
- Hypervigilance

{What is a key barrier to forming a therapeutic alliance in trauma therapy for survivors with self-blame? }

|type=&quot;()&quot;}

- Excessive emotional expression
- Over-reliance on external attributions
- Inability to recall traumatic events
+ Lack of trust in the therapist

&lt;/quiz&gt;
&lt;/div&gt;
{{Robelbox-close}}

==Therapeutic approaches to addressing self-blame==
{{Robelbox|width=30|theme={{{theme|3}}}|title=Case study: Sarah's story: Part 3}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;

After much encouragement from her sister, Sarah begins to see a trauma-focused therapist. Her therapist identifies that her overwhelming guilt is rooted in characterological self-blame, attributing the accident to a personal flaw. Through the use of cognitive processing therapy, Sarah learns to reinterpret the events and reconstruct a more balanced narrative, as well as recognising the limits of her control and foster self-compassion as a path to emotional recovery.
&lt;/div&gt;
{{Robelbox/close|theme=6}}

=== Cognitive and narrative intervention ===
[[File:Dr.Areej Khataybeh.jpg|174x174px|thumb|'''Figure 3'''. Example of a therapeutic approach to address self-blame and trauma.|right]]

Cognitive and narrative approaches to trauma recovery offer powerful tools for reshaping the internal landscape of survivors. Cognitive Processing Therapy (CPT) helps individuals identify and reframe distorted beliefs, particularly those rooted in self-blame, by challenging maladaptive thought patterns and replacing them with more balanced interpretations (Institute for Quality and Efficiency in Health Care, 2022). See Figure 3 for an example of a therapeutic setting in which interventions such as CPT may occur. This process disrupts entrenched cognitive loops that reinforce shame and helplessness. [[w:Narrative exposure therapy|Narrative exposure therapy]] (NET) builds on this by guiding survivors to construct a coherent life story that integrates traumatic events within a broader autobiographical context (Elbert, 2022). This not only aids identity reconstruction but also reduces the emotional intensity of fragmented memories (Schauer, 2015). [[w:Exposure therapy|Exposure therapy]], often used alongside these methods, directly addresses avoidance behaviours by gradually confronting trauma-related stimuli in a safe environment, helping to desensitize fear responses (Rubenstein, 2024). Together, these interventions promote meaning-making, allowing survivors to reinterpret their experiences through a lens of resilience rather than victimhood. A key therapeutic goal is to help individuals integrate new, compassionate self-narratives, shifting from internalized blame to a more empowered sense of self (Gilbert, 2014). This narrative reframing fosters emotional clarity and strengthens the survivor’s capacity to engage with life beyond the trauma.

=== Compassion-based and mindfulness therapies ===
Complementing cognitive strategies, compassion-based and mindfulness therapies offer a somatic and emotional pathway to healing (Fraser, 2024). Self-compassion interventions, such as [[w:Compassion-focused therapy|compassion-focused therapy]] (CFT), are particularly effective in reducing shame and self-criticism, common barriers to [[trauma]] recovery. These practices cultivate a nurturing internal voice that counteracts the harsh inner critic often amplified by traumatic experiences. [[w:Mindfulness-based stress reduction|mindfulness-based stress reduction]] (MBSR) enhances emotional regulation by helping individuals observe their thoughts and feelings without judgment, fostering a sense of safety and presence (Robins, 2012). [[w:Acceptance and commitment therapy|Acceptance and commitment therapy]] (ACT) further supports healing by teaching clients to distance from distressing thoughts, allowing them to act in alignment with personal values rather than trauma-driven fears (Wharton, 2019). [[w:Grounding (psychology)|Grounding]] techniques, such as breathwork and sensory anchoring, reconnect survivors with their bodies, counteracting dissociation and hyperarousal (Berberat, 2023). [[w:Somatic experiencing|Somatic experiencing]], a body-centred approach, helps release stored trauma by gently guiding individuals to notice and discharge physical tension linked to past events (Brom, 2017). These therapies not only soothe the nervous system but also rebuild trust in one’s bodily sensations and emotional responses. By integrating both cognitive and compassionate modalities, survivors can reclaim agency, restore relational intimacy, and move toward a more integrated and hopeful sense of self.

==Conclusion==
Self-blame remains one of the most pervasive and complex barriers to trauma recovery, often rooted in both emotional and cognitive mechanisms that reinforce its persistence. Survivors may internalise blame for traumatic events as a means to feel some control over events, or even gain power back when the reality was chaotic and without capacity. This style of attribution, which at first seems protective, might easily progress into a perpetual cycle of guilt, shame, and self-blame that spans psychological, emotional, and relational domains. Psychologically, self-blame is connected to increased risk of depression, anxiety, and PTSD, while emotionally it drives dysregulation and undermines sense of self-worth (Slanbekova, 2019). Relationally, it may result in people drifting apart from another person, not trusting them, and having an inability to develop secure relationships. Healing for self-blame is personalised and multi-modal in nature, aiming to address the cause of self-blame through cognitive restructuring, narrative reframing, and somatic regulation (Watt, 2011). Therapies such as CPT, CFT, and somatic experiencing provide complementary pathways to interrupt self-blame loops and return the self to a more balanced self-concept. Protective factor[ such as resilience, social support, and meaning-making, can serve as buffers against long-term damage (Ozbay, 2007). Resilience is not a static quality but something you cultivate through practice, engagement, community, and working through the therapeutic alliance. Using these insights, clinicians can guide interventions that honour the survivor’s particular environment and facilitate healing on multiple fronts. 

Looking ahead, new interventions offer hope to increase access and efficacy in treatment recovery. Digital CBT platforms, for instance, allow survivors to engage in structured cognitive work remotely, while peer-delivered support models leverage lived experience to build trust and cut down on stigmatisation (Zhang, 2023). This kind of innovation can be particularly valuable in underserved or marginalised communities. We need to delve further into the cultural dimensions of self-blame because ideas of responsibility, shame, and healing can vary enormously from culture to culture or spiritual contexts. Identity and the role of intersectionality, including race, gender, sexuality, and socioeconomic status must be at the heart of understanding trauma and its processes. Longitudinal research that follows recovery trajectories over time may help illustrate what types of interventions yield sustained benefits and how aspects like resilience and social support change (Janson, 2024). Incorporating neuroscience insights especially studies of neural correlates of self-blame and emotional regulation with trauma-informed policy can inform institutional practices in healthcare, education, and justice systems. For example neurofeedback and fMRI-based interventions are being evaluated to specifically address self-blame-specific brain activity in depression (Fennema, 2023). Ultimately, an integrated, holistic, inclusive, and evidence-based framework will be the best way forward for us to transform trauma care and equip survivors to reclaim their narrative.

{{Robelbox|theme=4|width=100%|title=Test your Knowledge !|iconwidth=48px|icon=Nuvola_apps_korganizer.svg}}&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
&lt;quiz display=simple&gt;
{ What is the primary goal of Cognitive Processing Therapy (CPT) in trauma recovery? }

|type=&quot;()&quot;}

+ To challenge distorted beliefs and replace them with balanced interpretations 
- To directly confront trauma-related stimuli
- To enhance emotional regulation through mindfulness
- To construct a coherent life story

{What is the relationship between resilience and trauma recovery? }

|type=&quot;()&quot;}

- Resilience is unrelated to trauma recovery
- Resilience is a fixed trait that cannot be cultivated
+ Resilience is dynamic process that can be developed through skill-building and community engagement 
- Resilience only applies to physical recovery

{What is a key research implication for understanding trauma recovery in diverse populations? }

|type=&quot;()&quot;}

- Limiting studies to short-term recovery outcomes
+ Exploring cultural dimensions of self-blame and healing
- Avoiding intersectionality in trauma research
- Focusing solely on individual experiences

&lt;/quiz&gt;
&lt;/div&gt;
{{Robelbox-close}}

==See also==
* [[Motivation and emotion/Book/2024/Attribution theory and emotion|Attribution theory and emotion]] (Book chapter, 2024)
* [[Helping Give Away Psychological Science/Coping with traumatic event|Coping with traumatic event]] (Wikiversity)
* [[Motivation and emotion/Book/2022/Psychological trauma|Psychological trauma]] (Book chapter, 2022)
* [[wikipedia:Psychological_trauma|Psychological trauma]] (Wikipedia)
* [[wikipedia:Self-blame_(psychology)|Self-blame (psychology)]] (Wikipedia)
* [[Social Skills/The Social Skill of Resilience|The social skill of resilience]] (Wikiversity)
* [[Motivation and emotion/Book/2023/Trauma and emotion|Trauma and emotion]] (Book chapter, 2023)

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{{Hanging indent|1=

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}}
==External links==
[https://www.sciencedirect.com/science/article/abs/pii/S0020138319306886 Psychological factors and recovery from trauma] (Science Direct)

[https://pmc.ncbi.nlm.nih.gov/articles/PMC9295847/ Self-Blame Attributions of Patients: a Systematic Review Study] (PubMed Central)

[https://journals.sagepub.com/doi/abs/10.1177/0146167296226002 Self-Blame Following a Traumatic Event: The Role of Perceived Avoidability] (Sage Journals)

[https://www.psychologytoday.com/us/blog/enlightened-living/201304/self-blame-the-ultimate-emotional-abuse Self Blame: The Ultimate Emotional Abuse] (Psychology today)

[https://www.sciencedirect.com/science/article/abs/pii/S0005796706000453 The role of self-blame for trauma as assessed by the Posttraumatic Cognitions Inventory (PTCI): A self-protective cognition?] (Science Direct)

[https://psychology.org.au/for-the-public/psychology-topics/trauma Trauma] (The Australian Psychological Society)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Self-blame]]
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{{title|Akrasia:&lt;br&gt;Why do people act against their better judgement?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
[[File: TikTok app.jpg |200px|thumb|right|alt=Temptation vs Self-Control| '''Figure 1'''. The digital temptation of tiktok]]

'''Imagine this ...'''

Sophie is a post-graduate student facing a looming thesis deadline. She knows she should do three hours of focused work, but when she sits down, she opens TikTok. 'Just one video' she tells herself, yet hours later, she's still scrolling, feeling guilty but unable to stop. She rationalises her behaviour as a necessary break despite recognising that it is the wrong decision and will set back her progress. This scenario clearly illustrates akrasia: acting against ones better judgement by giving into immediate temptation at the expense of long-term goals and what one knows is best for them.

{{RoundBoxBottom}}

*The problem with akrasia
*The importance of understanding akrasia and its psychological mechanisms
*How psychological science can help

{{RoundBoxTop|theme=3}}
'''Focus questions'''
* How have philosophers explained the conflict between reason and desire in human behaviour? 
* What factors lead people to act against their better judgement?  
* How does akrasia appear in everyday life today, what are the personal impacts?
* What methods or interventions can help individuals resist temptation and improve self-control? 
* In what ways might reconsidering akrasia change how we approach personal development and behaviour change? 
{{RoundBoxBottom}}

==Philosophical Foundations of Akrasia==

* Origins of the term [[w:Akrasia|akrasia]] in [[w:_Ancient_Greek_Philosophy|ancient greek philosophy]] (Steward, 1998)
* Early debates on the origins of wrongdoing
* Establishing groundwork for studying motivation in psychology

=== Socrates denial and Aristotle's Acceptance ===

*[[w:_Socrates|Socrates]]’ view on akrasia: Humans will always act according to what they truly know to be best; therefore, akrasia is impossible.
*[[w:Aristotle|Aristotle]]'s counterargument: akrasia results from weakness of will or lack of self-control.
*Legacy of the socrates, aristotle debate

=== From Moral Philosophy to Cognitive Science ===

* Changed from an ethical issue to a study of behavioural and cognitive processes (Bella, 2023)

&lt;quiz display=&quot;simple&quot;&gt;
'''Philosophical Foundations of Akrasia Quiz'''&lt;br&gt;
Aristotle believed that akrasia occurs when people act against their better judgment due to weakness of will.  
|type=&quot;()&quot;}
+ True
- False
&lt;/quiz&gt;

==Psychological Mechanisms of Self-Control Failure ==
Akrasia or the tendency to act against one's better judgement can be explained through understanding psychological mechanisms of self control failure. This self-control failure is driven by cognitive factors, emotional and motivation influences and behavioural patterns that weaken the link between intention and action. These psychological mechanisms seek to explain why people struggle to follow through on rational, intentional actions even though they are motivated to achieve long term goals.  

=== Cognitive Factors ===
* [[w:_Temporal_Discounting|Temporal discounting]] and [[w:_Present_Bias|present bias]]: immediate reward and comfort are often prioritised over long term goals (Kang &amp; Ikeda, 2016)
* Working memory, attention and planning affect self control (Oberauer, 2019)
* Cognitive overload and distractions

=== Emotional and Motivational Influence ===
* [[w:_Hot-Cold_Empathy_Gap|Hot-Cold empathy gaps]]: people misjudge how future emotional or physical states will influence their behaviour (Loewenstein, 2005)
* Strong emotions overpower rational decisions
* Motivation fluctuations

=== Behavioural Patterns ===
* Intention-Action gap: failing to act on a goal one intends to pursue (Faries, 2016)
* Habits can reinforce or undermine self-control (Stojanovic &amp; Wood, 2024)
* Environmental cues can trigger impulsive behaviours (Perry et al., 2014)

==Modern Contexts and Consequences==
* how akrasia manifests in everyday life today, why it seems more prevalent than ever, and the personal impacts

=== Digital Temptations ===
* Social media, streaming services and constant notifications promote instant gratification (Du et al., 2019)
* Easy access to online distractions increases procrastination and decreases self-control (Nadarajan et al., 2023)
* Digital environments increase impulsive behaviour, decreasing attention to long term goals (Wallace et al., 2023)

=== Health and Lifestyle Choices ===
* Short term desires can override long term health and fitness goals (Middleton et al., 2013)
* How psychological mechanisms influence lifestyle decision
* Chronic health behaviours often reflect repeated self-control failures

=== Financial and Career Decision ===
* How akrasia affects finances and career development
* Short-term temptations can undermine long-term financial or professional goals
* Tangible consequences of self-control failures in everyday life

==Practical Strategies for managing Acrasia==
* Evidence based cognitive, behavioural and environmental strategies for managing acrasia

=== '''Behavioural strategies''' ===
* Break tasks into smaller, manageable steps to reduce procrastination.
* Use cues, reminders, and structured routines to support consistent action
* Implement reward systems to reinforce desired behaviours

=== '''Cognitive strategies''' ===
* Mental contrasting: anticipate obstacles and plan concrete responses
* Reframe goals to enhance motivation and personal relevance.
* Visualisation and imagery techniques to strengthen commitment and follow-through.

=== '''Environmental and social strategies''' ===
* Minimise exposure to temptations in the physical and digital environment.
* Seek social support or accountability partners to maintain focus.
* Arrange surroundings and schedules to promote goal-consistent behaviour.

{{anchor|Feature box}}
;

==Conclusion==
150 to 330 words
*Nature of acrasia
*Key insights from philosophy and psychology
*Key messages: self control failure and akrasia are predictable and manageable, not moral failure.
*Understanding the psychology behind akrasia helps design better personal strategies.

==See also==
* [[Motivation and emotion/Book/2013/Self-control in health behaviours|Self-control and health behaviours]] (Book chapter, 2013)

== References ==
{{Hanging indent|1=

Bella, A. F. (2023). Psychological underpinnings of akrasia: A new integrative framework based on self-regulation vulnerabilities and failures. ''New Ideas in Psychology'', ''70'', 101027. &lt;nowiki&gt;https://doi.org/10.1016/j.newideapsych.2023.101027&lt;/nowiki&gt;

Du, J., Kerkhof, P., &amp; van Koningsbruggen, G. M. (2019). Predictors of Social Media Self-Control Failure: Immediate Gratifications, Habitual Checking, Ubiquity, and Notifications. ''Cyberpsychology, Behavior, and Social Networking'', ''22''(7), 477–485. &lt;nowiki&gt;https://doi.org/10.1089/cyber.2018.0730&lt;/nowiki&gt;

Faries, M. D. (2016). Why We Don’t “Just Do It.” ''American Journal of Lifestyle Medicine'', ''10''(5), 322–329. &lt;nowiki&gt;https://doi.org/10.1177/1559827616638017&lt;/nowiki&gt;

Kang, M.-I., &amp; Ikeda, S. (2016). Time discounting, present biases, and health-related behaviors: Evidence from Japan. ''Economics &amp; Human Biology'', ''21''(6), 122–136. &lt;nowiki&gt;https://doi.org/10.1016/j.ehb.2015.09.005&lt;/nowiki&gt;

Loewenstein, G. (2005). Hot-cold empathy gaps and medical decision making. ''Health Psychology'', ''24''(4, Suppl), S49–S56. &lt;nowiki&gt;https://doi.org/10.1037/0278-6133.24.4.s49&lt;/nowiki&gt;

Middleton, K. R., Anton, S. D., &amp; Perri, M. G. (2013). Long-Term Adherence to Health Behavior Change. ''American Journal of Lifestyle Medicine'', ''7''(6), 395–404. &lt;nowiki&gt;https://doi.org/10.1177/1559827613488867&lt;/nowiki&gt;

Nadarajan, S., Hengudomsub, P., &amp; Wacharasin, C. (2023). The role of academic procrastination on Internet addiction among Thai university students: A cross-sectional study. ''Belitung Nursing Journal'', ''9''(4), 384–390. &lt;nowiki&gt;https://doi.org/10.33546/bnj.2755&lt;/nowiki&gt;

Oberauer, K. (2019). Working Memory and Attention – A Conceptual Analysis and Review. ''Journal of Cognition'', ''2''(1). &lt;nowiki&gt;https://doi.org/10.5334/joc.58&lt;/nowiki&gt;

Perry, C. J., Zbukvic, I., Kim, J. H., &amp; Lawrence, A. J. (2014). Role of cues and contexts on drug-seeking behaviour. ''British Journal of Pharmacology'', ''171''(20), 4636–4672. &lt;nowiki&gt;https://doi.org/10.1111/bph.12735&lt;/nowiki&gt;

Steward, H. (1998). Akrasia. ''Routledge Encyclopedia of Philosophy'', ''3''(1). &lt;nowiki&gt;https://doi.org/10.4324/9780415249126-v003-1&lt;/nowiki&gt;

Stojanovic, M., &amp; Wood, W. (2024). Beyond Deliberate Self-Control: Habits Automatically Achieve Long-Term Goals. ''Current Opinion in Psychology'', ''42'', 101880–101880. &lt;nowiki&gt;https://doi.org/10.1016/j.copsyc.2024.101880&lt;/nowiki&gt;

Wallace, J., Boers, E., Ouellet, J., Afzali, M., &amp; Conrod, P. (2023). Screen time, impulsivity, neuropsychological functions and their relationship to growth in adolescent attention-deficit/ hyperactivity disorder symptoms. ''Scientific Reports'', ''13''(18108). &lt;nowiki&gt;https://doi.org/10.1038/s41598-023-44105-7&lt;/nowiki&gt;

}}

==External links==
* [https://www.theguardian.com/wellness/2024/mar/21/why-we-do-things-bad-for-us-impulse-habits-akrasia Why do we do things that are bad for us? The ancient philosophers had an answer] (The Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Motivation]]
[[Category:Motivation and emotion/Book/Self-control]]</text>
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      <timestamp>2026-06-11T10:08:53Z</timestamp>
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        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>Jtneill moved page [[Motivation and emotion/Book/2026/Akrasia and self-control failure]] to [[Motivation and emotion/Book/2026/Akrasia]] without leaving a redirect</comment>
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{{title|Akrasia:&lt;br&gt;Why do people act against their better judgement?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=3}}
[[File: TikTok app.jpg |200px|thumb|right|alt=Temptation vs Self-Control| '''Figure 1'''. The digital temptation of tiktok]]

'''Imagine this ...'''

Sophie is a post-graduate student facing a looming thesis deadline. She knows she should do three hours of focused work, but when she sits down, she opens TikTok. 'Just one video' she tells herself, yet hours later, she's still scrolling, feeling guilty but unable to stop. She rationalises her behaviour as a necessary break despite recognising that it is the wrong decision and will set back her progress. This scenario clearly illustrates akrasia: acting against ones better judgement by giving into immediate temptation at the expense of long-term goals and what one knows is best for them.

{{RoundBoxBottom}}

*The problem with akrasia
*The importance of understanding akrasia and its psychological mechanisms
*How psychological science can help

{{RoundBoxTop|theme=3}}
'''Focus questions'''
* How have philosophers explained the conflict between reason and desire in human behaviour? 
* What factors lead people to act against their better judgement?  
* How does akrasia appear in everyday life today, what are the personal impacts?
* What methods or interventions can help individuals resist temptation and improve self-control? 
* In what ways might reconsidering akrasia change how we approach personal development and behaviour change? 
{{RoundBoxBottom}}

==Philosophical Foundations of Akrasia==

* Origins of the term [[w:Akrasia|akrasia]] in [[w:_Ancient_Greek_Philosophy|ancient greek philosophy]] (Steward, 1998)
* Early debates on the origins of wrongdoing
* Establishing groundwork for studying motivation in psychology

=== Socrates denial and Aristotle's Acceptance ===

*[[w:_Socrates|Socrates]]’ view on akrasia: Humans will always act according to what they truly know to be best; therefore, akrasia is impossible.
*[[w:Aristotle|Aristotle]]'s counterargument: akrasia results from weakness of will or lack of self-control.
*Legacy of the socrates, aristotle debate

=== From Moral Philosophy to Cognitive Science ===

* Changed from an ethical issue to a study of behavioural and cognitive processes (Bella, 2023)

&lt;quiz display=&quot;simple&quot;&gt;
'''Philosophical Foundations of Akrasia Quiz'''&lt;br&gt;
Aristotle believed that akrasia occurs when people act against their better judgment due to weakness of will.  
|type=&quot;()&quot;}
+ True
- False
&lt;/quiz&gt;

==Psychological Mechanisms of Self-Control Failure ==
Akrasia or the tendency to act against one's better judgement can be explained through understanding psychological mechanisms of self control failure. This self-control failure is driven by cognitive factors, emotional and motivation influences and behavioural patterns that weaken the link between intention and action. These psychological mechanisms seek to explain why people struggle to follow through on rational, intentional actions even though they are motivated to achieve long term goals.  

=== Cognitive Factors ===
* [[w:_Temporal_Discounting|Temporal discounting]] and [[w:_Present_Bias|present bias]]: immediate reward and comfort are often prioritised over long term goals (Kang &amp; Ikeda, 2016)
* Working memory, attention and planning affect self control (Oberauer, 2019)
* Cognitive overload and distractions

=== Emotional and Motivational Influence ===
* [[w:_Hot-Cold_Empathy_Gap|Hot-Cold empathy gaps]]: people misjudge how future emotional or physical states will influence their behaviour (Loewenstein, 2005)
* Strong emotions overpower rational decisions
* Motivation fluctuations

=== Behavioural Patterns ===
* Intention-Action gap: failing to act on a goal one intends to pursue (Faries, 2016)
* Habits can reinforce or undermine self-control (Stojanovic &amp; Wood, 2024)
* Environmental cues can trigger impulsive behaviours (Perry et al., 2014)

==Modern Contexts and Consequences==
* how akrasia manifests in everyday life today, why it seems more prevalent than ever, and the personal impacts

=== Digital Temptations ===
* Social media, streaming services and constant notifications promote instant gratification (Du et al., 2019)
* Easy access to online distractions increases procrastination and decreases self-control (Nadarajan et al., 2023)
* Digital environments increase impulsive behaviour, decreasing attention to long term goals (Wallace et al., 2023)

=== Health and Lifestyle Choices ===
* Short term desires can override long term health and fitness goals (Middleton et al., 2013)
* How psychological mechanisms influence lifestyle decision
* Chronic health behaviours often reflect repeated self-control failures

=== Financial and Career Decision ===
* How akrasia affects finances and career development
* Short-term temptations can undermine long-term financial or professional goals
* Tangible consequences of self-control failures in everyday life

==Practical Strategies for managing Acrasia==
* Evidence based cognitive, behavioural and environmental strategies for managing acrasia

=== '''Behavioural strategies''' ===
* Break tasks into smaller, manageable steps to reduce procrastination.
* Use cues, reminders, and structured routines to support consistent action
* Implement reward systems to reinforce desired behaviours

=== '''Cognitive strategies''' ===
* Mental contrasting: anticipate obstacles and plan concrete responses
* Reframe goals to enhance motivation and personal relevance.
* Visualisation and imagery techniques to strengthen commitment and follow-through.

=== '''Environmental and social strategies''' ===
* Minimise exposure to temptations in the physical and digital environment.
* Seek social support or accountability partners to maintain focus.
* Arrange surroundings and schedules to promote goal-consistent behaviour.

{{anchor|Feature box}}
;

==Conclusion==
150 to 330 words
*Nature of acrasia
*Key insights from philosophy and psychology
*Key messages: self control failure and akrasia are predictable and manageable, not moral failure.
*Understanding the psychology behind akrasia helps design better personal strategies.

==See also==
* [[Motivation and emotion/Book/2013/Self-control in health behaviours|Self-control and health behaviours]] (Book chapter, 2013)

== References ==
{{Hanging indent|1=

Bella, A. F. (2023). Psychological underpinnings of akrasia: A new integrative framework based on self-regulation vulnerabilities and failures. ''New Ideas in Psychology'', ''70'', 101027. &lt;nowiki&gt;https://doi.org/10.1016/j.newideapsych.2023.101027&lt;/nowiki&gt;

Du, J., Kerkhof, P., &amp; van Koningsbruggen, G. M. (2019). Predictors of Social Media Self-Control Failure: Immediate Gratifications, Habitual Checking, Ubiquity, and Notifications. ''Cyberpsychology, Behavior, and Social Networking'', ''22''(7), 477–485. &lt;nowiki&gt;https://doi.org/10.1089/cyber.2018.0730&lt;/nowiki&gt;

Faries, M. D. (2016). Why We Don’t “Just Do It.” ''American Journal of Lifestyle Medicine'', ''10''(5), 322–329. &lt;nowiki&gt;https://doi.org/10.1177/1559827616638017&lt;/nowiki&gt;

Kang, M.-I., &amp; Ikeda, S. (2016). Time discounting, present biases, and health-related behaviors: Evidence from Japan. ''Economics &amp; Human Biology'', ''21''(6), 122–136. &lt;nowiki&gt;https://doi.org/10.1016/j.ehb.2015.09.005&lt;/nowiki&gt;

Loewenstein, G. (2005). Hot-cold empathy gaps and medical decision making. ''Health Psychology'', ''24''(4, Suppl), S49–S56. &lt;nowiki&gt;https://doi.org/10.1037/0278-6133.24.4.s49&lt;/nowiki&gt;

Middleton, K. R., Anton, S. D., &amp; Perri, M. G. (2013). Long-Term Adherence to Health Behavior Change. ''American Journal of Lifestyle Medicine'', ''7''(6), 395–404. &lt;nowiki&gt;https://doi.org/10.1177/1559827613488867&lt;/nowiki&gt;

Nadarajan, S., Hengudomsub, P., &amp; Wacharasin, C. (2023). The role of academic procrastination on Internet addiction among Thai university students: A cross-sectional study. ''Belitung Nursing Journal'', ''9''(4), 384–390. &lt;nowiki&gt;https://doi.org/10.33546/bnj.2755&lt;/nowiki&gt;

Oberauer, K. (2019). Working Memory and Attention – A Conceptual Analysis and Review. ''Journal of Cognition'', ''2''(1). &lt;nowiki&gt;https://doi.org/10.5334/joc.58&lt;/nowiki&gt;

Perry, C. J., Zbukvic, I., Kim, J. H., &amp; Lawrence, A. J. (2014). Role of cues and contexts on drug-seeking behaviour. ''British Journal of Pharmacology'', ''171''(20), 4636–4672. &lt;nowiki&gt;https://doi.org/10.1111/bph.12735&lt;/nowiki&gt;

Steward, H. (1998). Akrasia. ''Routledge Encyclopedia of Philosophy'', ''3''(1). &lt;nowiki&gt;https://doi.org/10.4324/9780415249126-v003-1&lt;/nowiki&gt;

Stojanovic, M., &amp; Wood, W. (2024). Beyond Deliberate Self-Control: Habits Automatically Achieve Long-Term Goals. ''Current Opinion in Psychology'', ''42'', 101880–101880. &lt;nowiki&gt;https://doi.org/10.1016/j.copsyc.2024.101880&lt;/nowiki&gt;

Wallace, J., Boers, E., Ouellet, J., Afzali, M., &amp; Conrod, P. (2023). Screen time, impulsivity, neuropsychological functions and their relationship to growth in adolescent attention-deficit/ hyperactivity disorder symptoms. ''Scientific Reports'', ''13''(18108). &lt;nowiki&gt;https://doi.org/10.1038/s41598-023-44105-7&lt;/nowiki&gt;

}}

==External links==
* [https://www.theguardian.com/wellness/2024/mar/21/why-we-do-things-bad-for-us-impulse-habits-akrasia Why do we do things that are bad for us? The ancient philosophers had an answer] (The Guardian)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Motivation]]
[[Category:Motivation and emotion/Book/Self-control]]</text>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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      <id>2815130</id>
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        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Motivation and emotion */ [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states?</comment>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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    <revision>
      <id>2815131</id>
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      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Motivation */ + [[/Feedback literacy/]] - What is feedback literacy and how can it be developed?</comment>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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    <revision>
      <id>2815135</id>
      <parentid>2815131</parentid>
      <timestamp>2026-06-10T22:27:59Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Emotion */ [[/Activism and emotion/]] - What emotions are involved in activism?</comment>
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      <text bytes="19976" sha1="7d4swk656uwl2asli71cxqd3a3es1b5" xml:space="preserve">{{/Banner}}
==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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    <revision>
      <id>2815139</id>
      <parentid>2815135</parentid>
      <timestamp>2026-06-10T22:32:28Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Motivation and emotion */ [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action?</comment>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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        <id>10242</id>
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      <comment>/* Emotion */ [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management?</comment>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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    <revision>
      <id>2815141</id>
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      <timestamp>2026-06-10T22:42:16Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
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      <comment>/* Emotion */ + [[/Human-robot trust/]] - What promotes and develops human trust of robots?</comment>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] – What is positive emotion dysregulation, what are its consequences, and what can be done about it? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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    <revision>
      <id>2815142</id>
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      <timestamp>2026-06-10T22:45:14Z</timestamp>
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        <id>10242</id>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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      <comment>/* Emotion */ + [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness?</comment>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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    <revision>
      <id>2815146</id>
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      <timestamp>2026-06-10T23:57:56Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Motivation */ + [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation?</comment>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness in the workplace/]] - How does cognitive hardiness protect against occupational stress and burnout? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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      <id>2815148</id>
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      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <minor />
      <comment>/* Emotion */</comment>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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      <timestamp>2026-06-11T00:18:16Z</timestamp>
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        <id>10242</id>
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      <comment>/* Emotion */ + [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions?</comment>
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      <text bytes="20871" sha1="5ayl1nvhmmjyh9g7s5k83p9ipy7vw64" xml:space="preserve">{{/Banner}}
==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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    <revision>
      <id>2815165</id>
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      <timestamp>2026-06-11T00:29:08Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>/* Motivation */ + [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours?</comment>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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      <comment>/* Motivation and emotion */ + [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love?</comment>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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        <id>10242</id>
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      <comment>/* Motivation */ [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine?</comment>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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      <id>2815172</id>
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        <id>10242</id>
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      <comment>/* Motivation */ [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives?</comment>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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      <comment>/* Emotion */ + [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it?</comment>
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==Motivation==
# [[/Akrasia and self-control failure/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and student motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity in goal striving/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

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==Motivation==
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Cost-benefit motivation and effort regulation/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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      <comment>/* Motivation */</comment>
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==Motivation==
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be developed and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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        <id>10242</id>
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==Motivation==
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Enhancing motivational dynamics in virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is getting started hard and how can this be overcome? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Mesolimbic pathway development and adolescent risk-taking/]] - How does maturation of reward circuits influence teenage sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention vs promotion mindset/]] - What are the motivational differences between focusing on safety versus growth? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How do unexpected financial gains influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect pursuit of social connection? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation and leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity versus abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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      <comment>/* Motivation */</comment>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Activism and emotion/]] - What emotions are involved in activism? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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    <revision>
      <id>2815200</id>
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        <id>10242</id>
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      <comment>/* Emotion */</comment>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does viewing one's body neutrally influence emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness help individuals cope with stress and adversity?  
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy and how does it differ from other forms of empathy? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, why do they matter, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - How does emotional flooding affect relationships and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How is emotional intelligence related to emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability versus strategy/]] – How do ability and strategy differ in shaping effective emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion in caregiving and helping roles? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotion of excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction mechanisms/]] - What psychological and neural processes underlie the extinction of fear responses? [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Fear of apocalypse/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human-robot trust/]] - What promotes and develops human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games]] - How do role-playing games allow players to explore their identity? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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      <comment>/* Emotion */</comment>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Laughing gas (nitrous oxide) and emotion/]] - How does nitrous oxide influence emotion? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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        <id>10242</id>
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      <comment>/* Emotion */</comment>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationship satisfaction/]] - How do love styles affect compatibility and long-term relationship outcomes? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Mood disorders and time experience/]] - How is time perceived differently in anxiety and depression? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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        <id>10242</id>
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      <comment>/* Emotion */</comment>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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    <revision>
      <id>2815219</id>
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        <id>10242</id>
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      <comment>/* Emotion */</comment>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie the experience of love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - What are the typical emotional responses to different types of noise? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence willingness to receive and act on feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and trust/]] - How does responsiveness influence the development and maintenance of trust? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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        <id>10242</id>
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      <comment>/* Emotion */</comment>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships influence the regulation of emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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      <comment>/* Emotion */</comment>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and emotion/]] - What are the affective aspects of wayfinding? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom versus interest/]] - How do interest and boredom interact to shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}}
# [[/Developing life purpose for well-being/]] - How does life purpose contribute to well-being and how can it be developed? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie the emergence of romantic love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|User Name}}
# [[/Indigenous Australian strengths-based psychology/]] - How do strengths-based approaches explain motivation and emotion among Indigenous Australians? {{ME-By|User Name}}
# [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|User Name}}
# [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|User Name}}
# [[/Moral emotions and moral motivation/]] - How do emotions such as moral anger, guilt, shame, and compassion motivate ethical behaviour, norm enforcement, and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|User Name}}
# [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|User Name}}
# [[/Moral emotions and ethical behaviour/]] - How do moral emotions motivate ethical and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|User Name}}
# [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|User Name}}
# [[/Moral emotions and ethical behaviour/]] - How do moral emotions motivate ethical and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How does discrepancy between expected and actual rewards influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – What is reinforcement sensitivity theory and how does it explain motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|User Name}}
# [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|User Name}}
# [[/Moral emotions and ethical behaviour/]] - How do moral emotions motivate ethical and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How does discrepancy between expected and actual rewards influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – How does reinforcement sensitivity theory explain individual differences in motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being framework/]] - How does the holistic social and emotional well-being model reframe Indigenous motivation and mental health? {{ME-By|User Name}}
# [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|User Name}}
# [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|User Name}}
# [[/Moral emotions and ethical behaviour/]] - How do moral emotions motivate ethical and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How does discrepancy between expected and actual rewards influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – How does reinforcement sensitivity theory explain individual differences in motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being in Indigenous Australians/]] - How does the holistic social and emotional well-being model reframe Indigenous Australian health and well-being? {{ME-By|User Name}}
# [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}}
# [[/Warm-glow giving, motivation, and emotion/]] - What are the motivational and emotional aspects of warm-glow giving? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|User Name}}
# [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|User Name}}
# [[/Moral emotions and ethical behaviour/]] - How do moral emotions motivate ethical and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How does discrepancy between expected and actual rewards influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – How does reinforcement sensitivity theory explain individual differences in motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being in Indigenous Australians/]] - How does the holistic social and emotional well-being model reframe Indigenous Australian health and well-being? {{ME-By|User Name}}
# [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}}
# [[/Warm-glow giving/]] - Why does giving feel good and how does this influence prosocial behaviour? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - What are the motivational-emotional aspects of wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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==Motivation==
# [[/Adolescent risk-taking and reward-system development/]] - How does reward circuit maturation influence adolescent sensation-seeking and impulsive behaviours? {{ME-By|User Name}}
# [[/Akrasia/]] - Why do people act against their better judgement? {{ME-By|User Name}}
# [[/Artificial intelligence and academic motivation/]] - How does artificial intelligence influence students’ motivation to learn, engage, and achieve? {{ME-By|User Name}}
# [[/Attachment styles and relatedness motivation/]] - How do attachment styles affect the need for relatedness? {{ME-By|User Name}}
# [[/Automaticity and goal pursuit/]] - How do habits and environmental cues drive unconscious goal pursuit? {{ME-By|User Name}}
# [[/Basal ganglia and motivation/]] - What is the role of the basal ganglia in motivated behaviour? {{ME-By|User Name}}
# [[/Charismatic leadership and follower motivation/]] - How does charismatic leadership inspire follower motivation? {{ME-By|User Name}}
# [[/Citizen science motivation/]] - What motivates participation in citizen science projects? {{ME-By|User Name}}
# [[/Competence motivation in self-determination theory/]] - How does the need for competence function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Consumer emotion measurement/]] - How can consumer emotion be measured? {{ME-By|User Name}}
# [[/Creative inspiration and effort/]] - How do inspiration and effort interact during the creative process? {{ME-By|User Name}}
# [[/Deliberative vs implemental mindset/]] - What are the motivational and cognitive differences between deliberative and implemental mindsets? {{ME-By|User Name}}
# [[/Developing a growth mindset/]] - How can a growth mindset be cultivated and sustained? {{ME-By|User Name}}
# [[/Dopamine and reward prediction/]] - How does dopamine affect the anticipation of rewards and subsequent emotional responses? {{ME-By|U3228742}}
# [[/Effort regulation and cost-benefit decision-making/]] - How is effort dynamically adjusted based on changing cost-benefit analysis during goal pursuit? {{ME-By|User Name}}
# [[/End-of-history illusion and motivation/]] - How does the EOHI influence motivation and what strategies mitigate its impact? {{ME-By|User Name}}
# [[/ERG theory and motivation/]] - What is Alderfer's ERG theory and how does it explain human motivation? {{ME-By|User Name}}
# [[/Epistemic motivation and the need for cognitive closure/]] - How does epistematic motivation and the need for cognitive closure influence our lives? {{ME-By|User Name}}
# [[/Extended process model of emotion regulation/]] – What is the extended process model and how does it explain how people regulate emotions? {{ME-By|User Name}}
# [[/Feedback literacy/]] - What is feedback literacy, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Fogg behaviour model/]] - How can the FBM be applied to understanding and changing behaviour? {{ME-By|User Name}}
# [[/Functional motives theory and environmental activism/]] - How does functional motives theory explain the motivations behind environmental activism? {{ME-By|User Name}}
# [[/Future orientation and criminal behaviour/]] - How does future orientation influence the risk of criminal activity? {{ME-By|User Name}}
# [[/Game of dice task and decision-making/]] - What does the game of dice task reveal about risk-based decision-making? {{ME-By|User Name}}
# [[/Gender and achievement motivation/]] - How does gender shape where, how, and under what conditions achievement motivation is expressed? {{ME-By|User Name}}
# [[/Generativity/]] - What is generativity and how does it impact behaviour and life outcomes? {{ME-By|User Name}}
# [[/Getting started/]] - Why is task initiation difficult and how to overcome it? {{ME-By|User Name}}
# [[/Goal striving dynamics/]] - What is the role of pushing and coasting in goal striving? {{ME-By|User Name}}
# [[/Hygiene motivation/]] - What motivates maintenance of personal hygiene? {{ME-By|User Name}}
# [[/Hypothalamus and homeostatic motivation/]] - How do hypothalamic circuits regulate hunger, thirst, and other survival-related motivations? {{ME-By|User Name}}
# [[/Impulsivity versus sensation-seeking/]] - What is the distinction between impulsivity and sensation-seeking and how does this affect behaviour? {{ME-By|User Name}}
# [[/Indigenous Australian role models and motivation/]] - How do role models influence aspirations, identity development, and motivation among Indigenous Australians? {{ME-By|User Name}}
# [[/Interrogation and compliance/]] - What psychological processes influence resistance and compliance during interrogation? {{ME-By|User Name}}
# [[/Investment model of commitment and social motivation/]] - How does the investment model of commitment relate to social motivation? {{ME-By|User Name}}
# [[/Lifelong learning motivation/]] - What motivates lifelong learning? {{ME-By|User Name}}
# [[/Machiavellian motivation/]] - What is the motivational role of Machiavellianism? {{ME-By|User Name}}
# [[/Mesolimbic pathway and addiction motivation/]] - What role does the ventral tegmental area to nucleus accumbens pathway play in addictive behaviours? {{ME-By|User Name}}
# [[/Metacognitive monitoring and productivity/]] - How does metacognitive monitoring influence goal attainment and productivity? {{ME-By|User Name}}
# [[/Mindsets and stigma/]] - What role do growth versus fixed mindsets play in prejudice and stigma? {{ME-By|User Name}}
# [[/Motivating virtual teams/]] – How can motivation in virtual teams be optimised? {{ME-By|User Name}}
# [[/Motivational effects of incarceration on Indigenous Australians/]] - What are the motivational effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Need to love and be loved/]] - How does the desire to give and receive love influence motivation? {{ME-By|User Name}}
# [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation be motivated and behaviour changed? {{ME-By|User Name}}
# [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? {{ME-By|User Name}}
# [[/Overconfidence in decision-making/]] - How does overconfidence bias affect judgement and decision-making? {{ME-By|User Name}}
# [[/Parental educational aspirations and student achievement/]] - How do parental aspirations shape children’s academic motivation and performance? {{ME-By|User Name}}
# [[/Parental motivations for homeschooling/]] - What motivates parents to homeschool their children? {{ME-By|User Name}}
# [[/Perfectionism and procrastination/]] - What is the role of perfectionism in procrastination and what can be done about it? {{ME-By|User Name}}
# [[/Pleasure anticipation and dopamine/]] - How does the brain's reward system generate motivation through expected rather than experienced pleasure? {{ME-By|User Name}}
# [[/Possible selves and goal pursuit/]] - How do possible selves influence motivation and goal-directed behaviour? {{ME-By|User Name}}
# [[/Power motivation in leadership/]] - How does power motivation influence leadership styles and effectiveness? {{ME-By|User Name}}
# [[/Prevention versus promotion mindset/]] - What are the motivational differences between prevention and promotion mindsets? {{ME-By|User Name}}
# [[/Protection motivation theory and environmental behaviour/]] - How does protection motivation theory explain engagement in pro-environmental behaviour? {{ME-By|User Name}}
# [[/Relatedness motivation in self-determination theory/]] - How does the need for relatedness function within self-determination theory to shape motivation and behaviour? {{ME-By|User Name}}
# [[/Role-play and communication skills training/]] - How does role-play facilitate the development of effective communication skills? {{ME-By|User Name}}
# [[/Scarcity versus abundance mindset/]] - How do scarcity and abundance mindsets develop and what are the motivational consequences? {{ME-By|User Name}}
# [[/Self-concept and motivation/]] - How does self-concept relate to motivation? {{ME-By|User Name}}
# [[/Self-determination theory and military veteran reintegration/]] - How do autonomy, competence, and relatedness shape psychological adjustment after military service? {{ME-By|User Name}}
# [[/Self-determination theory and physical activity/]] - How do autonomy, competence, and relatedness predict engagement in physical activity and exercise adherence? {{ME-By|User Name}}
# [[/Self-determination theory and social media use/]] - How do basic psychological needs explain patterns of social media engagement? {{ME-By|User Name}}
# [[/Sensation-seeking and dopamine/]] - What is the neurobiological relationship between sensation-seeking and dopamine? {{ME-By|User Name}}
# [[/Sex differences in sexual arousal patterns/]] - How do patterns of sexual arousal differ between males and females? {{ME-By|User Name}}
# [[/Sex work motivation/]] - What motivates sex work and how does this impact worker experiences? {{ME-By|User Name}}
# [[/Social dominance and power motivation/]] - What is the relationship between social dominance and power motivation? {{ME-By|User Name}}
# [[/Subcortical structures and motivational drive/]] - How do subcortical brain regions generate basic motivational impulses and energy? {{ME-By|User Name}}
# [[/Sun exposure and protection motivation/]] - What motivates sun exposure and protection behaviours? {{ME-By|User Name}}
# [[/Surrender motivation/]] - What is the motivational state of surrender and what are its impacts? {{ME-By|User Name}}
# [[/Thermoregulation and motivation/]] - How does the drive to maintain body temperature influence behaviour? {{ME-By|User Name}}
# [[/Tonic-phasic model of dopamine regulation/]] - What is the tonic/phasic model of dopamine regulation and how does affect behaviour? {{ME-By|User Name}}
# [[/Types of impulsivity/]] - What are the different types of impulsivity and how do they affect motivation? {{ME-By|User Name}}
# [[/Value congruence and motivation/]] - How does alignment between personal and situational values influence motivation? {{ME-By|User Name}}
# [[/Windfall gain effect/]] - How doe unexpected wealth influence behaviour and decision-making? {{ME-By|User Name}}
# [[/Youth environmental activism motivation/]] - What motivates young people to engage in environmental activism? {{ME-By|User Name}}

==Emotion==
# [[/Active versus passive social media use/]] - How do different patterns of social media engagement influence emotions and psychological wellbeing? {{ME-By|User Name}}
# [[/Affect heuristic/]] - What is the affect heuristic and how does it influence decision making? {{ME-By|User Name}}
# [[/Alcohol use for emotion regulation/]] - Why and how do people use alcohol to regulate their emotions? {{ME-By|User Name}}
# [[/Apocalyptic fear/]] - What is apocalyptic fear, what are its consequences, and how can it be dealt with? {{ME-By|User Name}}
# [[/Awe and the diminished self/]] - How does awe diminish the self and how can this be applied? {{ME-By|User Name}}
# [[/Awe and nature/]] - What is the relationship between awe and nature? {{ME-By|User Name}}
# [[/Biofeedback and emotion regulation/]] - How does biofeedback help individuals monitor and regulate their emotional states? {{ME-By|User Name}}
# [[/Body neutrality and emotional well-being/]] - How does a body-neutral perspective affect emotional well-being? {{ME-By|User Name}}
# [[/Breathing exercises and relaxation/]] - How can breathing exercises promote relaxation? {{ME-By|User Name}}
# [[/Cancer screening and emotion/]] - How do emotions such as fear, anxiety, and relief influence cancer screening uptake? {{ME-By|User Name}}
# [[/Cognitive hardiness and stress resilience/]] – How does cognitive hardiness promote resilience to stress and adversity? {{ME-By|User Name}}
# [[/Cognitive versus affective empathy/]] - What are the differences between cognitive and affective empathy and how do they contribute to prosociality? {{ME-By|User Name}}
# [[/Dark empathy/]] - What is dark empathy, what are its consequences, and what can be done to address it? {{ME-By|User Name}}
# [[/Dreams and emotional problem-solving/]] - How do REM dreams contribute to emotional processing and adaptive coping? {{ME-By|User Name}}
# [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? {{ME-By|User Name}}
# [[/Eco-emotions/]] - What are eco-emotions, how do they influence behaviour, and how can they be managed? {{ME-By|User Name}}
# [[/Emotional effects of incarceration on Indigenous Australians/]] - What are the emotional effects of incarcertation on Indigenous Australians?{{ME-By|User Name}}
# [[/Emotional expressivity/]] – What is emotional expressivity, why does it matter, and how can it be developed? {{ME-By|User Name}}
# [[/Emotional flooding in relationships/]] - Why does emotional flooding occur, how does it affect relationships, and what can be done about it? {{ME-By|User Name}}
# [[/Emotional intelligence and emotional wellbeing/]] - How does emotional intelligence affect emotional wellbeing? {{ME-By|User Name}}
# [[/Emotional role-playing/]] - How does role-playing influence emotional experience, expression, and regulation? {{ME-By|User Name}}
# [[/Emotion detection using artificial intelligence/]] - How can emotion be detected using artificial intelligence? {{ME-By|User Name}}
# [[/Emotion dysregulation/]] – What is emotion dysregulation, what are its consequences, and how can it be managed? {{ME-By|User Name}}
# [[/Emotion regulation ability and strategy/]] – How do ability and strategy differ in shaping emotion regulation? {{ME-By|User Name}}
# [[/Emotion regulation through exercise/]] - How do people use exercise to regulate their emotional states? {{ME-By|User Name}}
# [[/Emotions in activism/]] - How do emotions motivate, shape, and sustain activism? {{ME-By|User Name}}
# [[/Empathy fatigue and emotional exhaustion/]] - How does sustained empathic engagement contribute to emotional exhaustion? {{ME-By|User Name}}
# [[/Enjoyment and learning/]] - How does enjoyment influence learning? {{ME-By|User Name}}
# [[/Environmental volunteering and wellbeing/]] - How does participation in environmental volunteering influence volunteers' subjective wellbeing? {{ME-By|User Name}}
# [[/Excitement as an emotion/]] - What is the emotional excitement and how does it influence behaviour and wellbeing? {{ME-By|User Name}}
# [[/Fear extinction/]] - What psychological and neural processes underlie the extinction of fear responses? {{ME-By|User Name}}
# [[/Focalism in affective forecasting/]] - What is focalism and how does it bias predictions about future emotional experiences? {{ME-By|User Name}}
# [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? {{ME-By|User Name}}
# [[/Human trust of robots/]] - What psychological factors shape human trust of robots? {{ME-By|User Name}}
# [[/Indigenous Australian funeral practices and grieving/]] - How do Indigenous Australian funeral practices assist with grieving? {{ME-By|User Name}}
# [[/Identify exploration through role-playing games/]] - How do role-playing games facilitate identity exploration and self-discovery? {{ME-By|User Name}}
# [[/Introjection and guilt-based motivation/]] - What role do shame and guilt play in introjected forms of behavioural regulation? {{ME-By|User Name}}
# [[/Love styles and relationships/]] - How do love styles influence relationship satisfaction and stability? {{ME-By|User Name}}
# [[/Melatonin and seasonal mood/]] - What role does melatonin play in seasonal mood changes? {{ME-By|User Name}}
# [[/Mindfulness and nature connectedness/]] - How does mindfulness influence nature connectedness? {{ME-By|User Name}}
# [[/Mood and cognitive performance/]] – How do different mood states impact attention, memory, and problem solving? {{ME-By|User Name}}
# [[/Moodiness/]] - What is moodiness, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Neurobiology of love/]] - What neural systems and biochemical processes underlie love? {{ME-By|User Name}}
# [[/Neurofeedback and emotional regulation/]] - How can neurofeedback influence enhance emotional regulation? {{ME-By|User Name}}
# [[/Nitrous oxide and emotion/]] - How does nitrous oxide influence emotional experience and mood? {{ME-By|User Name}}
# [[/Noise and emotion/]] - How do different types of noise affect emotional experience and wellbeing? {{ME-By|User Name}}
# [[/Opponent process theory and emotion/]] - What role do opposing affective states play in emotional experience? {{ME-By|User Name}}
# [[/Phubbing and emotion/]] - What are the emotional causes and consequences of phubbing? {{ME-By|User Name}}
# [[/Positive emotion dysregulation/]] - What is positive emotion dysregulation and how does it affect psychological functioning? {{ME-By|User Name}}
# [[/Psychological preparation for natural disasters/]] - How can people psychologically prepare for natural disasters? {{ME-By|User Name}}
# [[/Psychological safety and feedback uptake/]] - How does psychological safety influence openness to feedback? {{ME-By|User Name}}
# [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? {{ME-By|Username}}
# [[/Remote work and well-being/]] - How does remote work influence employee well-being? {{ME-By|Username}}
# [[/Responsiveness and interpersonal trust/]] - How does responsiveness foster trust in relationships? {{ME-By|User Name}}
# [[/Romantic jealousy/]] - Why does romantic jealousy occur, what are its impacts, and how can it be managed? {{ME-By|User Name}}
# [[/Secondary trauma in healthcare workers/]] - What are the emotional consequences of secondary trauma in healthcare settings? {{ME-By|User Name}}
# [[/Seasonal affective disorder/]] - What is SAD, why does it occur, and how can it be managed? {{ME-By|User Name}}
# [[/Self-blame and emotion/]] – How does self-blame influence emotional responses to negative events? {{ME-By|User Name}}
# [[/Self-disclosure and emotional intimacy/]] – How does self-disclosure foster emotional closeness in relationships? {{ME-By|User Name}}
# [[/Self-stigma and emotion/]] - How does self-stigma impact emotional well-being? {{ME-By|User Name}}
# [[/Social connection and emotion regulation/]] - How do social relationships help people emotions? {{ME-By|User Name}}
# [[/Socioemotional selectivity theory and wellbeing in ageing/]] - How do social and emotional experiences affect wellbeing as people age? {{ME-By|User Name}}
# [[/Spirituality and resilience/]] - What is the relationship between spirituality and psychological resilience? {{ME-By|User Name}}
# [[/Subjective wellbeing homeostasis theory/]] - How does homeostatic theory explain the stability and regulation of subjective wellbeing? {{ME-By|User Name}}
# [[/Technology-based pain management/]] - How can technology-based tools alter pain perception and pain management? {{ME-By|User Name}}
# [[/Theory of positive disintegration and personal growth/]] - What is the TPD and how can it be applied to personal growth? {{ME-By|User Name}}
# [[/Time perception in mood disorders/]] - How do anxiety and depression alter the subjective experience of time? {{ME-By|User Name}}
# [[/Trust in artificial intelligence/]] - What psychological factors shape human trust of artificial intelligence systems? {{ME-By|User Name}}
# [[/Trust rebuilding after trauma/]] - How can trauma survivors develop trust in similar situations again? {{ME-By|User Name}}
# [[/Volunteer wellbeing/]] - How does volunteering affect volunteer's subjective wellbeing? {{ME-By|User Name}}
# [[/Wayfinding and affective experience/]] - How do emotions influence navigation and spatial behaviour? {{ME-By|User Name}}

==Motivation and emotion==
# [[/Boredom and interest/]] - How do boredom and interest shape emotional and motivational states? {{ME-By|User Name}}
# [[/Falling in love/]] - What motivational and emotional processes underlie romantic attraction and falling in love? {{ME-By|User Name}}
# [[/Life purpose and well-being/]] - How does a sense of purpose contribute to well-being and how can it be cultivated? {{ME-By|User Name}}
# [[/Moral emotions and ethical behaviour/]] - How do moral emotions motivate ethical and prosocial action? {{ME-By|User Name}}
# [[/Oxytocin as a neuromodulator/]] - What are the motivational and emotional effects of oxytocin as a neuromodulator? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How does discrepancy between expected and actual rewards influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Reinforcement sensitivity theory/]] – How does reinforcement sensitivity theory explain individual differences in motivation and emotion? {{ME-By|User Name}}
# [[/Reward prediction error/]] - How do reward prediction errors influence learning, emotion, and motivation? {{ME-By|User Name}}
# [[/Social and emotional well-being in Indigenous Australians/]] - How does the holistic social and emotional well-being model reframe Indigenous Australian health and well-being? {{ME-By|User Name}}
# [[/Strengths-based Indigenous Australian psychology/]] - How can strengths-based perspectives enhance understanding of Indigenous motivation and emotion? {{ME-By|User Name}}
# [[/Warm-glow giving/]] - Why does giving feel good and how does this influence prosocial behaviour? {{ME-By|User Name}}
# [[/Wisdom, motivation, and emotion/]] - How do motivational and emotional processes contribute to wisdom? {{ME-By|User Name}}

[[Category:Motivation and emotion/Book]]</text>
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    <title>Talk:Motivation and emotion/Book/2026/Akrasia</title>
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      <comment>Jtneill moved page [[Talk:Motivation and emotion/Book/2026/Akrasia and self-control failure]] to [[Talk:Motivation and emotion/Book/2026/Akrasia]] without leaving a redirect</comment>
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|-
| [[File:Crystal Clear app ktip.svg|48px|left]]
| {{#if:|Hi [[User:{{{1}}}|{{{1}}}]].|}} FYI, the recommended [[Wikiversity]] heading style uses [[w:Letter case#Sentence_case|sentence casing]]. For example:&lt;br&gt;
&lt;big&gt;&lt;big&gt;Self-determination theory&lt;/big&gt;&lt;/big&gt;
rather than
&lt;big&gt;&lt;big&gt;Self-Determination Theory&lt;/big&gt;&lt;/big&gt;

Here's an example chapter with correct heading casing: [[Motivation and emotion/Book/2019/Growth mindset development|Growth mindset development]]

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:21, 23 August 2025 (UTC)
|}
&lt;!-- Official topic development feedback --&gt;
{{METF/2025
|1=
&lt;!-- Title --&gt;
# Title and sub-title correctly worded and use [[w:Letter case#Sentence casing|sentence casing]]
|2=
&lt;!-- Headings --&gt;
# See earlier comment about [[#heading casing|heading casing]]
&lt;!-- Heading structure --&gt;
# Well developed 2-level heading structure. Meaningful headings clearly relate directly to the core topic.
&lt;!-- Alignment with focus questions --&gt;
# Very good alignment between sub-title, focus questions, and heading structure, but there may be room for improvement
&lt;!-- Other ---&gt;
# Use default heading formatting (i.e., avoid additional formatting such as bold, italics, underline, changing the size etc.)
&lt;!-- GenAI ---&gt;
# Are the headings based on [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
|3=
&lt;!-- Overview--&gt;
# Excellent – Scenario, image, evocative description of the problem/topic, and focus questions
&lt;!-- Scenario --&gt;
# A scenario or case study is presented in a feature box with an image at the start of this section
&lt;!-- Description --&gt;
# A basic description of the problem/topic is planned or presented
&lt;!-- Focus questions --&gt;
# Good alignment between focus questions and heading structure, but consider closer alignment
|4=
&lt;!-- Key points--&gt;
&lt;!-- Overall --&gt;
# Excellent – key points are well developed for each section
&lt;!-- Scope --&gt;
# It may be that all planned aspects cannot be reasonably covered within the final word count, so be selective and concentrate on key aspects that address the question in the sub-title. For example, concentrate on the psychological rather than philosophical aspects.
&lt;!-- Theory and research --&gt;
# Reasonably good coverage of theory; strive to balance the theoretical content with critical review of relevant research
&lt;!-- Citations --&gt;
# Promising use of citations
&lt;!-- Other --&gt;
# ''Avoid providing too much background information''. Aim to briefly summarise general concepts and provide internal links to relevant book chapters and/or Wikipedia pages for further information. Focus most of the chapter on ''directly answering the core question(s)'' posed by the chapter sub-title.
# Use correct capitalisation ([https://apastyle.apa.org/style-grammar-guidelines/capitalization APA style is a &quot;down&quot; style]) – [https://polishedpaper.com/blog/capitalization-apa-style more info]
# Use [https://www.abc.net.au/education/learn-english/australian-vs-american-spelling/11244196 Australian spelling] (e.g., analyze → analyse; behavior → behaviour)
&lt;!-- GenAI ---&gt;
# Do these key points include [[Motivation and emotion/Assessment/Using generative AI|genAI content]]? If so, this needs to be acknowledged in the edit summaries, otherwise it violates academic integrity.
&lt;!-- Conclusion --&gt;
# Conclusion is underway
|5=
&lt;!-- Figure --&gt;
# Relevant figure(s) are presented and captioned
&lt;!-- Caption --&gt;
# Figure caption(s) provide(s) a somewhat clear description that is connected with the main text, but could be improved
# Figure caption(s) should include '''Figure X'''. ...
&lt;!-- Cite --&gt;
# Cite each figure at least once in the main text using APA style (e.g., see Figure 1)
|6=
&lt;!-- Learning feature --&gt;
&lt;!-- Interwiki links ---&gt;
# Excellent in-text [[m:Help:Interwiki linking|interwiki links]] for first mention of key terms to [[w:|Wikipedia]] and/or [[Motivation and emotion/Book|book chapters]]
&lt;!-- Scenarios/examples/case studies --&gt;
# Consider use of more scenarios/examples/case studies 
&lt;!-- Quiz --&gt;
# Promising use of quiz question(s)
# Place quiz each question in the most relevant section
# Focus the quiz question(s) on the take-home messages
&lt;!-- Tables --&gt;
# Also consider using tables to summarise key information
|7=
&lt;!-- References --&gt;
&lt;!-- Overall --&gt;
# Good
&lt;!-- Systematic reviews --&gt;
# What are the most relevant systematic reviews/meta-analyses about this topic?
&lt;!-- APA style --&gt;
# Check and correct [https://apastyle.apa.org/instructional-aids/reference-guide.pdf APA referencing style]:
## capitalisation
## [[Help:Wikitext quick reference|italicisation]]
## make doi hyperlinks active (i.e., clickable)
|8=
&lt;!-- Resources --&gt;
&lt;!-- See also --&gt;
# See also
## One of two link types provided
### Also link to relevant [[w:|Wikipedia]] pages
&lt;!-- External links --&gt;
# External links
## One of two required external links provided
|9=
&lt;!-- User page --&gt;
# Used effectively
&lt;!-- Description about self --&gt;
# Excellent description about self provided
&lt;!-- Links to profile(s) --&gt;
# Consider linking to your [https://portfolio.canberra.edu.au/ eportfolio] page and/or any other professional online profile or resume such as [https://www.linkedin.com/ LinkedIn]. This is not required, but it can be useful to interlink your professional networks.
&lt;!-- Link to book chapter --&gt;
# A link to the book chapter is provided
|10=
&lt;!-- Social contribution --&gt;
# Excellent – at least three different types of contributions with direct link(s) to evidence
}}
-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 21:21, 23 August 2025 (UTC)

== Sentence casing ==

Hi Gaby, I've just made a small edit in your overview section where I removed the capital letters of some words. APA 7th formatting recommend sentence casing as an overall guide. This includes words like akrasia, cognitive load, and others, which should all be written with lowercase letters unless its the start of the sentence. For example: Cognitive behavioural therapy (CBT) is... This doesn't apply for names or other proper nouns :). I'd recommend looking through the rest of your outline and making those adjustments.
The chapter is looking great! Keep it up [[User:Lachlancanning04|Lachlancanning04]] ([[User talk:Lachlancanning04|discuss]] • [[Special:Contributions/Lachlancanning04|contribs]]) 02:10, 29 August 2025 (UTC)

== Conclusion  ==

Hi Gaby,

You’ve done a good job highlighting the main points in your conclusion, especially that akrasia and self-control failures are predictable and manageable rather than just moral failings, and the link to personal strategy shows some practical, down-to-earth thinking.
That said, the ending is a bit short and comes across more like a set of notes than a fully rounded final paragraph. While the key ideas are there, they don’t quite flow together into a coherent story, so the section feels a bit patchy. The conclusion would feel more complete and impactful if it included a few examples of how understanding akrasia could help with everyday decision-making or goal-setting. [[User:Dsanad|Dsanad]] ([[User talk:Dsanad|discuss]] • [[Special:Contributions/Dsanad|contribs]]) 10:30, 15 November 2025 (UTC)</text>
      <sha1>70xcbeust5pcbf942wt5cx6785l9u7s</sha1>
    </revision>
  </page>
  <page>
    <title>Just sustainability transitions: a living review</title>
    <ns>0</ns>
    <id>326060</id>
    <revision>
      <id>2815205</id>
      <parentid>2814557</parentid>
      <timestamp>2026-06-11T10:41:20Z</timestamp>
      <contributor>
        <username>Jeanne Noiraud</username>
        <id>1366702</id>
      </contributor>
      <comment>/* Modelling knowledge */</comment>
      <origin>2815205</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="59404" sha1="6nz4mae16ku8jj2p15ucuhob6z5ak27" xml:space="preserve">== Contributors ==
{| class=&quot;wikitable&quot;
|+
!Name
!Affiliation
!ORCID
!Contribution
|-
|Adélie Ranville
|IAE de Grenoble, CERAG lab (https://ror.org/0509qp208)
|https://orcid.org/0000-0002-3993-6135
|Research design, database search, article screening, knowledge modelling
|-
|Amélie Pereira
|
|
|Meta-data enrichement
|-
|
|
|
|
|}
Contribution statistics are visible here : https://xtools.wmcloud.org/pageinfo/en.wikiversity.org/Just_sustainability_transitions:_a_living_review

== Introduction ==

=== Definition of living review ===
The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked&lt;ref name=&quot;Why1&quot;&gt;{{Cite Q |Q40040379 }}&lt;/ref&gt;. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot;&gt;{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}&lt;/ref&gt;&lt;ref name=&quot;:6&quot;&gt;{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}&lt;/ref&gt;. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication&lt;ref name=&quot;Why1&quot;/&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;. Living systematic reviews exists to address this common problem&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/&lt;/ref&gt;. It is therefore particularly useful in rapidly evolving fields of research&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;, such as just transition. 

[[wikidata:Q33002955|Knowledge graphs]], a structured representation of knowledge in the form of a graph, linked together by relationships that encode explicit meanings between these entities, are very suitable for conducting living systematic reviews&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;Fotopoulou&quot;&gt;{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.&lt;/ref&gt;. Advances in AI could render certain older methodological types of living systematic reviews obsoletes&lt;ref&gt;{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}&lt;/ref&gt;, as IA are useful to extract, filter and classify datas&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;. [[Large language models]] (LLM) are &quot;on the rise&quot; (2025), but &quot;not yet ready for use&quot;&lt;ref&gt;{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}&lt;/ref&gt;.

The living review method relevant for just transition because it includes topic such as energy democracy which necessitate transdisciplinarity and consolidation of fragmented literature&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;.

===  Definitions of just transition : ===
* «a fair and equitable process of moving towards a post-carbon society’. »&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}&lt;/ref&gt;.


The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions&lt;ref name=&quot;:0&quot; /&gt;. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions&lt;ref name=&quot;:0&quot; /&gt;. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research &amp; Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}&lt;/ref&gt; and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 
 
=== Definition of Procedural justice ===
Procedural justice is about the fairness of decision-making processes related to transitions&lt;ref name=&quot;:4&quot; /&gt; such as the inclusion of those impacted by these decisions&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}&lt;/ref&gt;. Procedural justice can include issues of community and citizen participation in decision making, their political representation their consultation or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities&lt;ref&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. For example, the participation of affected communities in decisions related to the construction of new infrastructures&lt;ref name=&quot;:0&quot; /&gt;.     

== Methodology ==

=== Wikidata and the semantic web ===&lt;!-- Add introduction to what wikidata is and how the triplet works in a pedagogical manner

Example of good description here : https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k#fig1
 --&gt;
&quot;A knowledge graph is a structured representation of knowledge that captures information in a machine-readable format.&lt;ref name=&quot;:9&quot;&gt;{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}&lt;/ref&gt; A knowledge graph consists of a graph or network of interconnected data points, where each data point represents a piece of information or a concept, and the relationships between them are explicitly defined. Knowledge graphs organize and store data in a format that facilitates information retrieval, data analysis, and reasoning.&quot;&lt;ref&gt;{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}&lt;/ref&gt;

== Building a corpus and enriching bibliographic metadata ==

=== Database search ===
We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. 
 
{| class=&quot;wikitable&quot;
|+
!Keywords search
!Database
!Search date
!Filters
!Number of results
|-
|(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) 
|Web of Science (all databases, all dates)
|December 2025
|Document type: Review Article
|362
|}

=== Article screening ===
Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were 

* Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...)
* Not literature reviews (e.g. review of policies, initiatives, cases, review notes, book review...)
* Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors or included justice consideration only in “future research” suggestions
* Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy
* Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper

=== Importing selected articles into Wikidata ===
To import the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in the database. Next we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the &quot;export to Wikidata QuickStatements&quot; function of Zotero and use the QuickStatements tool to add them to Wikidata.

Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well  because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|&quot;Merge&quot; gadget]] on Wikidata. We checked manually for duplicated statments in those items.

=== Article classification through meta-data enrichement ===&lt;!-- Add : What is meta-data enrichement --&gt;

Existing review try to classify existing articles according to various criteria such as industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt).&lt;ref name=&quot;:5&quot; /&gt; We selected the most relevant properties in Wikidata to reflect these classifications : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe its geographical focus.

==== Main subjects ====
We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : 
{| class=&quot;wikitable&quot;
|+
!Qid
!Main topic
!Description
|-
|[[d:Q42377797|Q42377797]]
|acceptability
|characteristic of a thing being subject to acceptance for some purpose
|-
|[[d:Q2798912|Q2798912]]
|accountability
|concept of responsibility in ethics, governance and decision-making
|-
|[[d:Q421953|Q421953]]
|actor–network theory
|theory within social science
|-
|[[d:Q84459973|Q84459973]]
|affordability
|
|-
|[[d:Q185836|Q185836]]
|age of a person
|time elapsed since a person was born
|-
|[[d:Q4764988|Q4764988]]
|animal studies
|field in which animals are studied in a variety of cross-disciplinary ways
|-
|[[d:Q4338318|Q4338318]]
|awareness
|state or ability to perceive, to feel, or to be conscious of events, objects, or sensory patterns
|-
|[[d:Q4930066|Q4930066]]
|blue carbon
|carbon captured by the world's coastal ocean ecosystems
|-
|[[d:Q430460|Q430460]]
|capability approach
|economic theory
|-
|[[d:Q7569|Q7569]]
|child
|human between birth and puberty
|-
|[[d:Q4116870|Q4116870]]
|civic engagement
|individual or group activity addressing issues of public concern
|-
|[[d:Q125928|Q125928]]
|climate change
|human-caused changes to climate on Earth
|-
|[[d:Q260607|Q260607]]
|climate change 
adaptation
|process of adjustment to actual or expected climate change and its effects, seeking to moderate or avoid harm or exploit beneficial opportunities
|-
|[[d:Q1291678|Q1291678]]
|climate justice
|term linking the climate crisis with environmental and social justice
|-
|[[d:Q2270945|Q2270945]]
|co-creation
|product or service design process in which input from consumers plays a central role
|-
|[[d:Q16972712|Q16972712]]
|co-design
|approach to design attempting to actively involve all stakeholders
|-
|[[d:Q16324410|Q16324410]]
|coproduction
|product or service design process in which input from consumers plays a central role
|-
|[[d:Q11024|Q11024]]
|communication
|act of conveying intended meaning
|-
|[[d:Q177634|Q177634]]
|community
|social unit of human organisms who share common values
|-
|[[d:Q5154673|Q5154673]]
|community choice aggregation
|alternative energy supply system
|-
|[[d:Q113514984|Q113514984]]
|community energy
|delivery of community-led renewable energy, energy demand reduction and energy supply projects
|-
|[[d:Q65807646|Q65807646]]
|community participation
|The taking part by members of a community in decisionmaking processes related to the development of their community
|-
|[[d:Q188843|Q188843]]
|cosmopolitanism
|ideology that all human beings belong to a single community, based on a shared morality
|-
|[[d:Q11693783|Q11693783]]
|decarbonization
|change of economy, especially of energy industries, towards lower carbon dioxide emissions
|-
|[[d:Q284289|Q284289]]
|deliberative democracy
|form of democracy focusing on consensus
|-
|[[d:Q7174|Q7174]]
|democracy
|form of government
|-
|[[d:Q552284|Q552284]]
|distributive justice
|concept of the socially just allocation of goods
|-
|[[d:Q1230584|Q1230584]]
|diversity
|concept in sociology and political studies
|-
|[[d:Q1049066|Q1049066]]
|ecological economics
|research field on the interdependence of human economies and natural ecosystems
|-
|[[d:Q8134|Q8134]]
|economics
|social science that studies the production, distribution, and consumption of goods and services
|-
|[[d:Q868575|Q868575]]
|empowerment
|providing increased autonomy
|-
|[[d:Q295865|Q295865]]
|ecosystem service
|benefits created by nature, forests and environmental systems
|-
|[[d:Q138359220|Q138359220]]
|energy citizenship
|involvement of citizens in energy-related decisions
|-
|[https://www.wikidata.org/w/index.php?title=Q131444737&amp;redirect=no Q131444737]
|community energy
|[redirection]
|-
|[[d:Q16869822|Q16869822]]
|energy consumption
|amount of energy or power used
|-
|[[d:Q1358789|Q1358789]]
|senior
|elderly person
|-
|[[d:Q14944319|Q14944319]]
|energy democracy
|concept in environmental justice movement
|-
|[[d:Q192704|Q192704]]
|energy efficiency
|ratio between the useful energy output and the input of a machine
|-
|[[d:Q24965464|Q24965464]]
|energy modeling
|process of building computer models of energy systems in order to analyze them
|-
|[[d:Q1805337|Q1805337]]
|energy policy
|policy addressing energy issues
|-
|[[d:Q1341244|Q1341244]]
|energy poverty
|lack of access to modern energy services
|-
|[[d:Q3406659|Q3406659]]
|energy production
|conversion of energy from a primary source into a form useful to humans
|-
|[[d:Q117091181|Q117091181]]
|energy justice
|subconcept of economic equality
|-
|[[d:Q3456219|Q3456219]]
|energy renovation
|building works aimed at reducing energy consumption and decarbonising the energy sources used
|-
|[[d:Q2700433|Q2700433]]
|energy security
|national security considerations of energy availability
|-
|[[d:Q837718|Q837718]]
|energy storage
|capture of energy produced at one time for use at a later time
|-
|[[d:Q795757|Q795757]]
|energy transition
|long-term structural change towards sustainable energy systems
|-
|[[d:Q1479527|Q1479527]]
|environmental justice
|system of fairness
|-
|[[d:Q771773|Q771773]]
|fairness
|concept in sociology and generally the interaction of society
|-
|[[d:Q56395513|Q56395513]]
|farming system
|method of agricultural production defined by its physical practices and economic characteristics
|-
|[[d:Q5465532|Q5465532]]
|food system
|all processes and infrastructure involved in feeding a population
|-
|[[d:Q4421|Q4421]]
|forest
|dense collection of trees covering a relatively large area
|-
|[[d:Q48277|Q48277]]
|gender
|social concept which distinguish the different gender categories
|-
|[[d:Q1553864|Q1553864]]
|governance
|all of the processes of governing, whether undertaken by a government, market or network, whether over a family, tribe, formal or informal organization or territory and whether through the laws, norms, power or language of an organized society
|-
|[[d:Q8458|Q8458]]
|human rights
|inalienable fundamental rights to which a person is inherently entitled
|-
|[[d:Q11376059|Q11376059]]
|human rights violation
|act or omission which contravene the principles of human rights
|-
|[[d:Q103817|Q103817]]
|indigenous people
|first inhabitants of an area and their descendants
|-
|[[d:Q113561794|Q113561794]]
|indigenous science
|indigenous knowledge applied to the scientific method
|-
|[[d:Q770480|Q770480]]
|injustice
|quality relating to unfairness or undeserved outcomes
|-
|[[d:Q17142211|Q17142211]]
|interactional justice
|the perceived appropriateness of interpersonal treatment
|-
|[[d:Q1516555|Q1516555]]
|intersectionnality
|theoretical framework of multidimensional oppression
|-
|[[d:Q6316391|Q6316391]]
|just transition
|Framework developed by the trade union movement to encompass wide range of social interventions needed to secure decent work opportunities and a greener economy.
|-
|[[d:Q366139|Q366139]]
|legitimation
|the process of making something acceptable and normative to a group
|-
|[[d:Q3027857|Q3027857]]
|living lab
|user-centered, open innovation ecosystem integrating research and innovation in real life communities
|-
|[[d:Q59679511|Q59679511]]
|low income
|home with little money
|-
|[[d:Q43619|Q43619]]
|natural environment
|all living and non-living things occurring naturally on Earth or some region thereof
|-
|[[d:Q127514833|Q127514833]]
|nature-positive
|global goal to halt and reverse nature loss by 2030
|-
|[[d:Q13023682|Q13023682]]
|non-human
|organism not in the genus Homo
|-
|[[d:Q728646|Q728646]]
|partnership
|arrangement in which parties agree to cooperate to advance their mutual interests
|-
|[[d:Q3907287|Q3907287]]
|policy making
|the act of developing policy
|-
|[[d:Q9357091|Q9357091]]
|political theory
|class of theory
|-
|[[d:Q265425|Q265425]]
|postcolonialism
|academic discipline
|-
|[[d:Q25107|Q25107]]
|power
|ability to influence the behavior of others
|-
|[[d:Q442100|Q442100]]
|procedural justice
|fairness in the processes that resolve disputes and allocate resources
|-
|[[d:Q7249406|Q7249406]]
|project governance
|management framework
|-
|[[d:Q7257735|Q7257735]]
|public engagement
|Policy-making practice
|-
|[[d:Q541936|Q541936]]
|public participation
|participation of citizens in various policy decisions and planning processes
|-
|[[d:Q6142016|Q6142016]]
|recognition justice
|social philosophy theory
|-
|[[d:Q10509953|Q10509953]]
|renewable electricity
|electricity from renweable sources
|-
|[[d:Q12705|Q12705]]
|renewable energy
|energy collected from renewable resources
|-
|[[d:Q56510941|Q56510941]]
|renewable energy policy
|
|-
|[[d:Q1165392|Q1165392]]
|restorative justice
|approach to justice where victims and perpetrators mediate a restitution agreement
|-
|[[d:Q4414036|Q4414036]]
|rural population
|inhabitants of rural areas or of small towns classified as rural
|-
|[[d:Q17152351|Q17152351]]
|smart system
|adaptive intelligent systems
|-
|[[d:Q187588|Q187588]]
|social class
|group of people categorized in a hierarchy based on socioeconomic factors
|-
|[[d:Q264892|Q264892]]
|social justice
|concept that discrimination recognized in society should be remedied
|-
|[[d:Q34749|Q34749]]
|social science
|academic disciplines concerned with society and the relationships between individuals in society
|-
|[[d:Q2930198|Q2930198]]
|stakeholder participation
|involvement of groups or individuals affected by the actions of an entity
|-
|[[d:Q125359881|Q125359881]]
|sustainability transition
|
|-
|[[d:Q219416|Q219416]]
|sustainability
|ability of human civilization to coexist with the biosphere in a steady state
|-
|[[d:Q131201|Q131201]]
|sustainable development 
|mode of human development that meets current demands without compromising the needs of future generations
|-
|[[d:Q7649586|Q7649586]]
|Sustainable Development Goals
|set of United Nations-defined global development goals and climate change
|-
|[[d:Q69883|Q69883]]
|urban planning
|technical and political process concerned with the use of land and design of the urban environment
|-
|[[d:Q920600|Q920600]]
|urban renewal
|program of land redevelopment in cities, often where there is urban decay
|-
|[[d:Q3376054|Q3376054]]
|vulnerable population
|group of persons whose range of options is severely limited, are subjected to coercion, or who may be compromised in their ability to give informed consent
|-
|[[d:Q107389921|Q107389921]]
|water-management
|
|-
|[[d:Q7981051|Q7981051]]
|well-being
|measure of how well life is to someone or a group with factors such as health, happiness and satisfaction
|-
|[[d:Q467|Q467]]
|woman
|female adult human
|-
|[[d:Q188867|Q188867]]
|future studies
|study of possible, probable, and preferable social, technological and political futures
|-
|[[d:Q1038171|Q1038171]]
|participatory design
|active involvement of all stakeholders in the design process
|}
&lt;!-- include all below items using the wikidata link template
 --&gt;

Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords.

==== Study types ====
Our review included only litterature reviews. We first read abstracts to identify all the [https://angryloki.github.io/wikidata-graph-builder/?item=Q2412849&amp;property=P279&amp;mode=reverse different types of litterature reviews] present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved these method items using the methodological references cited in the reviewed papers. The types of reviews were : 
{| class=&quot;wikitable&quot;
|+
!Qid
!Study type
!Description
|-
|[[d:Q603441|Q603441]]
|bibliometrics
|statistical analysis of written publications, such as books or articles
|-
|[[d:Q472342|Q472342]]
|scientometrics
|study of measuring and analysing science, technology and innovation
|-
|[[d:Q815382|Q815382]]
|meta-analysis
|statistical method that summarizes data from multiple sources
|-
|[[d:Q1504425|Q1504425]]
|systematic review
|publication type, study that gathers, analyzes, and communicates the results of research and information on a topic
|-
|[[d:Q2412849|Q2412849]]
|literature review
|process of information search and text of a review article (Q7318358), which includes the current knowledge including substantive findings, as well as theoretical and methodological contributions to a particular topic
|-
|[[d:Q6822263|Q6822263]]
|meta-regression
|statistical tool used in meta-analyses
|-
|[[d:Q7301211|Q7301211]]
|realist evaluation
|[...]
|-
|[[d:Q17007303|Q17007303]]
|combinatorial meta-analysis 
|[...]
|-
|[[d:Q70470634|Q70470634]]
|network meta-analysis
|meta-analysis of randomized trials in which estimates of comparative treatment effects are visualized and interpreted from a network of interventions
|-
|[[d:Q101116078|Q101116078]]
|scoping review
|search for concepts by mapping the language and data which surrounds those concepts and adjusting the search method iteratively to synthesize evidence and assess the scope of an area of inquiry
|-
|[[d:Q110665014|Q110665014]]
|narrative review
|type of literature review, without structured method of retrieval and analysis
|-
|[[d:Q137174203|Q137174203]]
|conceptual review
|academic research aiming to review existing concepts and definitions in the litterature
|-
|[[d:Q137174450|Q137174450]]
|critical review
|type of literature review analysing strenghts, major contributions, mistakes and neglected issues in an academic field of research
|-
|[[d:Q137209848|Q137209848]]
|integrative literature review
|type of literature review
|-
|[[d:Q110665014|Q137211242]]
|narrative review
|type of literature review, without structured method of retrieval and analysis
|}Then, we added the {{Wikidata entity link|P8363}} of each articles based on the abstract and method sections. In case of doubt, we compared our interpretation.

==== Research site ====
When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article &quot;{{Wikidata entity link|Q137901202}}&quot; focused on {{Wikidata entity link|Q132959}}.

==== Results ====
The table listing all the papers in the sample can be visualized [https://tabernacle.toolforge.org/?#/tab/manual/Q137211155%0A%0A%0A%0A%0A%0A%0AQ114306483%0A%0A%0A%0A%0AQ137901181%0A%0A%0A%0AQ137901182%0A%0A%0A%0A%0A%0A%0A%0AQ137901183%0A%0A%0AQ114306476%0A%0A%0A%0A%0AQ137901184%0A%0A%0A%0A%0AQ137901185%0A%0A%0A%0A%0A%0AQ137901186%0A%0A%0A%0A%0A%0A%0AQ137901187%0A%0A%0A%0A%0A%0A%0AQ137901188%0A%0A%0A%0A%0AQ137210566%0A%0A%0A%0A%0AQ114306511%0A%0A%0A%0A%0A%0AQ137901191%0A%0A%0A%0A%0AQ137901192%0A%0A%0A%0A%0AQ137901193%0A%0A%0A%0A%0AQ135979013%0A%0A%0A%0A%0A%0A%0A%0AQ137901195%0A%0A%0A%0A%0A%0AQ137901196%0A%0A%0A%0A%0A%0A%0AQ137901197%0A%0A%0A%0A%0AQ136447761%0A%0A%0A%0AQ137901199%0A%0A%0A%0A%0A%0A%0AQ129652515%0A%0A%0A%0A%0A%0A%0AQ137901201%0A%0A%0A%0A%0A%0AQ137901202%0A%0A%0A%0A%0AQ137901203%0A%0A%0A%0AQ137901204%0A%0A%0A%0A%0A%0A%0A%0AQ137901205%0A%0A%0A%0A%0AQ137901206%0A%0A%0A%0A%0A%0A%0A%0A%0AQ137901207%0A%0A%0A%0A%0AQ129203992%0A%0A%0A%0A%0A%0A%0AQ114197507%0A%0A%0A%0AQ137901161%0A%0A%0A%0A%0A%0A%0A%0AQ137901209%0A%0A%0A%0A%0A%0AQ137901210%0A%0A%0A%0A%0A%0AQ137901211%0A%0A%0A%0A%0AQ11420462%0A%0AQ137901213%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0AQ104887325%0A%0A%0A%0A%0A%0AQ137901162%0A%0A%0AQ137901163%0A%0A%0A%0A%0AQ137901164%0A%0A%0A%0A%0A%0AQ137901215%0A%0A%0A%0A%0AQ137901216%0A%0A%0A%0A%0A%0A%0A%0A%0AQ137901217%0A%0A%0A%0A%0AQ115448818%0A%0A%0A%0A%0AQ137901218%0A%0A%0A%0AQ137901219%0A%0A%0A%0A%0AQ137901220%0A%0A%0A%0A%0A%0AQ137901221%0A%0A%0A%0A%0A%0AQ137901222%0A%0A%0A%0A%0AQ137901223%0A%0A%0AQ137901224%0A%0A%0A%0AQ137901225%0A%0A%0A%0A%0A%0A%0AQ137901226%0A%0A%0A%0AQ137901227%0A%0A%0AQ137901182/Len%3BP921%3BP6153%3BP8363%3BP50 here] (be careful if you are logged into Wikidata as the table is editable).

== Modelling knowledge ==
Concept maps can be a powerful literature review tool&lt;ref&gt;{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}&lt;/ref&gt; allowing to synthetize theoretical statements about relationship between concepts&lt;ref&gt;{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}&lt;/ref&gt;. In the present study, we explored how concept map can be used to model the knowledge present in the paper we selected. 

[define knowledge modelling]

==== Wikidata ontology ====
Wikidata &quot;supports multiple coexisting classification&quot; and allow multiple ontological frameworks to coexist.&lt;ref name=&quot;:8&quot;&gt;{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}&lt;/ref&gt;

It also supports epistemic pluralism : different worldviews can be represented in wikidata, even though scientific knowledge is preferred.&lt;ref name=&quot;:8&quot; /&gt;

See more on membership properties : https://www.wikidata.org/wiki/Help:Basic_membership_properties

See the discussion on cause modelling : https://www.wikidata.org/wiki/Help:Modeling_causes/en

==== Conceptual modelling ====
We first reflected on what kind of wikidata properties could be used to represent concepts and theories in wikidata. Capturing the content of a concept is not straightforward and there are various approaches coming from psychology and philosophy on the matter&lt;ref&gt;{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}&lt;/ref&gt; we summarize these approaches below and examine which wikidata properties exist to represent them.
* Definition: the content of a concept can be formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}...
* Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}}.
* Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}.
* Essence: the content of a concept is &quot;something&quot; deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can create it before really knowing what all its properties will be.
* Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}.

==== Categorization and conceptualisation practices in management sciences ====
In management sciences « systematic categorizing is the best and perhaps only method for clearing up semantic confusion, management scholars never take the classical approaches to categorizing that facilitated tremendous progress in the physical sciences, and seldomly build on extant categorial schemes. »&lt;ref&gt;{{Cite journal|last=Pierce|first=Jason R.|date=2025-01|title=Categorizing Concepts and Phenomena in Management Research: A Four-Phase Integrative Review and Recommendations|url=http://journals.aom.org/doi/full/10.5465/annals.2023.0052|journal=Academy of Management Annals|language=en|volume=19|issue=1|page=28|pages=9–37|doi=10.5465/annals.2023.0052|issn=1941-6520}}&lt;/ref&gt;.

Some scholars discussed how conceptualization should be done&lt;ref&gt;{{Cite journal|last=Podsakoff|first=Philip M.|last2=MacKenzie|first2=Scott B.|last3=Podsakoff|first3=Nathan P.|date=2016-04|title=Recommendations for Creating Better Concept Definitions in the Organizational, Behavioral, and Social Sciences|url=https://journals.sagepub.com/doi/10.1177/1094428115624965|journal=Organizational Research Methods|language=en|volume=19|issue=2|pages=159–203|doi=10.1177/1094428115624965|issn=1094-4281}}&lt;/ref&gt;,&lt;ref&gt;{{Cite journal|last=Makowski|first=Piotr Tomasz|date=2021-10|title=Optimizing Concepts: Conceptual Engineering in the Field of Management—The Case of Routines Research|url=http://journals.aom.org/doi/full/10.5465/amr.2019.0252|journal=Academy of Management Review|language=en|volume=46|issue=4|pages=702–724|doi=10.5465/amr.2019.0252|issn=0363-7425}}&lt;/ref&gt;.

==== Thematic networks ====
[[File:Thematic network example.jpg|thumb|547x547px|Structure of a thematic network (Source: Attride-Stirling 2001)]]
A thematic network is “simply a way of organizing a thematic analysis of qualitative data”&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}&lt;/ref&gt;. It is compatible with classical coding strategies such as [[grounded theory]]&lt;ref&gt;{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}&lt;/ref&gt;. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data&lt;ref name=&quot;:7&quot; /&gt;. The principle is to assemble basic themes into more general themes.   

Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding (e.g. grounded theory) to identify themes and sub-themes.   

However, the nature of the relationship between these various themes and sub-themes is often not specified.  

*

==== Causal networks ====
The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers sometime present models with boxes and arrows representing correlations and/or causations&lt;ref&gt;{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}&lt;/ref&gt;. In qualitative research, building grounded theory models is about &quot;[accounting] for not only all the major emergent concepts, themes, and dimensions, but also for their dynamic interrelationships. Speaking in classic boxes-and-arrows terms, this process amounts to assembling the constellation of boxes with a special focus on the arrows.&quot;&lt;ref&gt;{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}&lt;/ref&gt; Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can &quot;feedback&quot; (two variables can influence each other)&lt;ref&gt;{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}&lt;/ref&gt;.

Wikidata includes several properties to describe causal relationships:
* {{Wikidata entity link|P828}}
* {{Wikidata entity link|P1542}}
* {{Wikidata entity link|P1537}}
* {{Wikidata entity link|P1479}} : it is difficult to identify single causes for social phenomenons, many factors having an effect on the subject item will likely be contributing factors

== Testing concept modelling on {{Wikidata entity link|Q14944319}} ==
We started by experimenting the modelling of concept by focusing on the concept of {{Wikidata entity link|Q14944319}}. We selected a subset of papers which had energy democracy as main topic :

* {{Wikidata entity link|Q137901202}}
* {{Wikidata entity link|Q137901196}}
* {{Wikidata entity link|Q137901182}}
* {{Wikidata entity link|Q136447761}}
* {{Wikidata entity link|Q129652515}}
* {{Wikidata entity link|Q114306483}}

We read each paper and used them as source to enter statements in the item {{Wikidata entity link|Q14944319}}. For example, &quot;Energy democracy is both an ideal and a process&quot;&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;, we thus entered the wikidata statement {{Wikidata entity link|Q14944319}} is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q840396}}, using the paper as source. The result of this first step is visible in the archival version of the item (22 May 2026) here https://www.wikidata.org/w/index.php?title=Q14944319&amp;oldid=2495982191.

Ontology challenges:
*{{Wikidata entity link|P31}}: concepts may have a dual nature because they designate at the same time an idea and the entity that this idea represent. Energy democracy is a concept, an ideal, a process and an outcome. 
*'''Process versus outcome :''' For material processes, the distinction between process and outcome is rather simple. For example, in Wikidata, {{Wikidata entity link|Q11629}} (practice of applying paint) is different from {{Wikidata entity link|Q3305213}} (visual artwork), and this distinction is based on the criterion &quot;{{Wikidata entity link|Q127270577}}&quot;. However, this distinction is less straightforward for social processes that do not have an end. Such processes are ongoing and outcomes cannot be separated as clearly.
* '''Ideal versus reality :''' Concepts do not have goals in themselves, but the reality they represent can have goals. To distinguish goals from the process to reach it, we used {{Wikidata entity link|P3712}} to describe ideals and {{Wikidata entity link|P2670}} to describe processes.
* '''Phenomenon versus theory :''' Wikidata current items are not really suited to model &quot;meta-research&quot; statements. For example, modelling the idea tha the literature on energy democracy is fragmented would require creating an item representing the energy democracy literature, not just energy democracy in general. Similarly, it can be difficult to model the chronological evolution of the definition of an idea (although it could be technically possible). It is hard to represent in Wikidata affirmations related to missing knowlege, propositions of untested hypothesis, critique of existing research or research agenda recommandations
* '''Origin of discourses versus origin of practices :''' To distinguish the causes of the concepts/discourses and the causes of the phenomenon itself, we used {{Wikidata entity link|P3938}} to indicate the origins of the concept or the movements promoting it.


Some of the statements we added may seem contradictory. However, Wikidata supports &quot;because statements essentially point to referenceable sources of information and different sources may provide contradicting information, it's possible to represent a plurality of perspectives on Wikidata&quot;&lt;ref&gt;{{Cite web|url=https://www.wikidata.org/wiki/Help:Statements#Plurality_and_consensus|title=Help:Statements - Wikidata|website=www.wikidata.org|language=en|access-date=2026-06-08}}&lt;/ref&gt;. The {{Wikidata entity link|Q14944319}} concept could be split into more precise concepts to distinguish the social movement advocating for it, the political concept theorizing it and the concrete initiatives implementing it. However, the current sources do not make this distinction for now. 

Other challenges
* Wikidata does not seem to be the best tool to model quantitative statements, for example, the paper {{Wikidata entity link|Q137901196}} states that &quot;9.8% of the final energy consumed in developing countries comes from modern renewable energy sources&quot;. Including energy data in Wikidata require using or creating specific properties (e.g. {{Wikidata entity link|P6826}})
* When concepts are not precisely defined, statements cannot be modelled correctly. For example, in the sentence &quot;management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served&quot;&lt;ref&gt;{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research &amp; Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}&lt;/ref&gt;, &quot;choice&quot; could refer to {{Wikidata entity link|Q111986453}}, {{Wikidata entity link|Q1331926}}, or {{Wikidata entity link|Q12888920}} as &quot;choice&quot; can refer to the availability of different options, or the decision process to chose among them.


Advantages : 

* Link toward unique identifiers for concepts, but also laws (e.g. {{Wikidata entity link|Q139764294}})

== Interactions with the Wikidata community ==

* Some Wikidata contributors added labels for {{Wikidata entity link|Q14944319}} in other languages such as Armenian or Slovenian.

== Data visualisation ==

=== Filter statements ===

* Visualize only statements using a specitic source. Example : https://w.wiki/PFqH
* Visualize only items which are part to the present project (require that all items of the project include the statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}).

=== Mapping a concept ===
Scholia request &quot;topic in context&quot;

=== Mapping sources consensus ===
Visualise graphs and use the number of references to determine edge thickness/weight.

== Writing ==
To cite articles we used the [[Template:Cite Q|Cite Q template.]] Each reference is an item in Wikidata and the template retrieve the necessary data to generate the citation references below.


== Future research ==
The analysis of knowledge graph could in theory allow to make logical deduction to generate new data&lt;ref name=&quot;:9&quot; /&gt;.

Reflect on the future of scholarly communication : https://hal.science/hal-03277615/file/OPERAS_Future_of_Scholarly_Communication_06.2021.pdf

== Data == 
{| class=&quot;wikitable sortable&quot;
! QID !! Year !! DOI !! Title
|-
| [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review
|-
| [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review
|-
| [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review
|-
| [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter?
|-
| [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset.
|-
| [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies?
|-
| [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection
|-
| [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development
|-
| [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research
|-
| [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition
|-
| [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning
|-
| [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review
|-
| [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view
|-
| [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory
|-
| [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries
|-
| [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review
|-
| [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions
|-
| [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies
|-
| [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes
|-
| [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation
|-
| [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives
|-
| [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies
|-
| [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda
|-
| [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice?
|-
| [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review
|-
| [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research
|-
| [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape
|-
| [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models
|-
| [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review
|-
| [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions
|-
| [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions
|-
| [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation
|-
| [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings
|-
| [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda
|-
| [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review
|-
| [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework
|-
| [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende
|-
| [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa
|-
| [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities
|-
| [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion
|-
| [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review
|-
| [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights
|-
| [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review
|-
| [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations
|-
| [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance
|-
| [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions
|-
| [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review
|-
| [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice
|-
| [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice
|-
| [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review
|-
| [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review
|-
| [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions
|-
| [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition
|-
| [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy
|-
| [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends
|-
| [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience
|}


== References ==
{{References}}</text>
      <sha1>6nz4mae16ku8jj2p15ucuhob6z5ak27</sha1>
    </revision>
  </page>
  <page>
    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
    <ns>2</ns>
    <id>326765</id>
    <revision>
      <id>2815037</id>
      <parentid>2814964</parentid>
      <timestamp>2026-06-10T14:03:08Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Complementary chord pairs */</comment>
      <origin>2815037</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="73765" sha1="iwjds3s5r0wj2pbz8h3m4yz5i0q47sx" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Great hexagon.png|100px]]
| rowspan=&quot;3&quot; |400 regular [[600-cell#Hexagons|great hexagons]]&lt;br&gt; (1200 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Hexagons and hexagrams|2{10/3}]]&lt;br&gt;#4
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

...

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="73692" sha1="r7s29sot8mzd9ldgog6j9evvaco9621" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a dodecahedron cell, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal dodecahedron cell at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

...

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>r7s29sot8mzd9ldgog6j9evvaco9621</sha1>
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    <revision>
      <id>2815045</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="73708" sha1="60hsgdkxfig0tu77ppgamqtyq5cwt85" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

...

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="73866" sha1="57tlwas0f4evmwst3r2gkzhi993s03p" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;]], every vertex is the apex of a polyhedral pyramid of edge length &lt;math&gt;c_{1\ge t \le 30}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

...

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>57tlwas0f4evmwst3r2gkzhi993s03p</sha1>
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      <id>2815047</id>
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        <username>Dc.samizdat</username>
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      <text bytes="73884" sha1="cjgpampeligrkwe95abpeoue32rqrta" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of a polyhedral pyramid of edge length &lt;math&gt;c_{t}&lt;/math&gt; for &lt;math&gt;1 \ge t \le 30&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

...

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>cjgpampeligrkwe95abpeoue32rqrta</sha1>
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      <id>2815048</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="73856" sha1="t7nzhahfk4a43u63nk6coma8ys51hqr" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of a 30 polyhedral pyramids of edge length &lt;math&gt;c_{t}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

...

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>t7nzhahfk4a43u63nk6coma8ys51hqr</sha1>
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    <revision>
      <id>2815049</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Complementary chord pairs */</comment>
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      <text bytes="73859" sha1="nb98g7vg0anmk7b3zjmbjs1y27ivn2u" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 30 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of a 30 polyhedral pyramids of edge length &lt;math&gt;c_{t}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
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|1.480~
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| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
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| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
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| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
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|{{radic|2}}
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|- style=&quot;background: seashell;&quot; |
|1.414~
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== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

...

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>nb98g7vg0anmk7b3zjmbjs1y27ivn2u</sha1>
    </revision>
    <revision>
      <id>2815050</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="73857" sha1="kcga3vbmw175v20n7axdy0ditaipcak" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 30 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 30 polyhedral pyramids of edge length &lt;math&gt;c_{t}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

...

== Conclusions ==

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

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>kcga3vbmw175v20n7axdy0ditaipcak</sha1>
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      <id>2815051</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="73857" sha1="n5yvoo4dd96tz7vabiuhvfbugz8ginm" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 polyhedral pyramids of edge length &lt;math&gt;c_{t}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>n5yvoo4dd96tz7vabiuhvfbugz8ginm</sha1>
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    <revision>
      <id>2815052</id>
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      <timestamp>2026-06-10T15:35:45Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <comment>/* Complementary chord pairs */</comment>
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      <text bytes="73908" sha1="lbo6mfv4c0g02gcqpn3l6te6ke0huvt" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius and edge length &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius and edge length &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of edge length &lt;math&gt;c_{t}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
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|1.480~
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|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
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| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
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| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
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|{{radic|2}}
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|- style=&quot;background: seashell;&quot; |
|1.414~
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|1.414~
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|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>lbo6mfv4c0g02gcqpn3l6te6ke0huvt</sha1>
    </revision>
    <revision>
      <id>2815053</id>
      <parentid>2815052</parentid>
      <timestamp>2026-06-10T15:42:10Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Complementary chord pairs */</comment>
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      <text bytes="73927" sha1="3s5zs0mho4pdlmkj56rwnymjxifpr0z" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius and edge length &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius and edge length &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of edge length &lt;math&gt;c_{t}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>3s5zs0mho4pdlmkj56rwnymjxifpr0z</sha1>
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        <username>Dc.samizdat</username>
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      <text bytes="74160" sha1="pk52zhz2sa58q8dzl4fnwdmbvx0up3w" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius and edge length &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius and edge length &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74136" sha1="61ztq78mvndj8jfw5ltxk1ctkb6xqps" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74198" sha1="kjcjsvwglg11p7uh13pbjxbnny6w327" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px| {30/15}=15{2} ]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great hexagons of the 120-cell.png|100px]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74180" sha1="k8179jgn7mvsoslu5dx6l0ildakii9i" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2} ]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:25.2° × 154.8° chords great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74180" sha1="2wqe24vv46ofggqql1gtni3wyt5ahje" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2} ]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Great decagon rectangle.png|100px]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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        <username>Dc.samizdat</username>
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      <text bytes="74197" sha1="cc638je7xqtumopci0q3im3an8m7c96" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2} ]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5 × √3.5 great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular great dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74220" sha1="e8h5nwrr5jvjcnz56027r2h6t2w7gbu" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2} ]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
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| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
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|{{radic|2.191~}}
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|- style=&quot;background: gainsboro;&quot; |
|1.345~
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|1.480~
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|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
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| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
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| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
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|{{radic|2}}
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|- style=&quot;background: seashell;&quot; |
|1.414~
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|1.414~
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|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <id>2815063</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="74222" sha1="flb7l46omnuxz9wairsx3teabdw7ker" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg&lt;br&gt;{30/15}=15{2}|100px]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74219" sha1="odefqc71plsakj10ps3xetswo38jkqz" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
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|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74225" sha1="8majd10ushwcr63njcqn842wj66bc0y" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|thumb|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74219" sha1="odefqc71plsakj10ps3xetswo38jkqz" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <id>2815067</id>
      <parentid>2815066</parentid>
      <timestamp>2026-06-10T18:03:56Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Complementary chord pairs and sections */</comment>
      <origin>2815067</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="74165" sha1="eov50syue6v020t7oj3rrkcetfkgur1" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Great pentagons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="74178" sha1="4qrnymxl8897xa62ickgl4smt1c4n69" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Great square rectangle.png|100px]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:30-gon#Triacontagram|{30/7}]]&lt;br&gt;#7
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74174" sha1="4xcrcqm2kuizjgvuxb30wm3f0jdarik" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74225" sha1="aipdyjnz6aj2r0m6g70mp8ubundcsyc" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[600-cell#Squares and octagrams|{24/5}]]&lt;br&gt;#9
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
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| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
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|{{radic|2.191~}}
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|- style=&quot;background: gainsboro;&quot; |
|1.345~
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|1.480~
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|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
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| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
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| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
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|{{radic|2}}
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|- style=&quot;background: seashell;&quot; |
|1.414~
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|1.414~
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|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>aipdyjnz6aj2r0m6g70mp8ubundcsyc</sha1>
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    <revision>
      <id>2815071</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="74230" sha1="cyzjco02hayxu7ols39wamwe0dh7cf6" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |5𝝅&lt;br&gt;[[600-cell#Decagons and pentadecagrams|{15/2}]]&lt;br&gt;#5
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74228" sha1="cdur6qinxyjp16lxp1d7qz5jre7cw9z" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
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|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{30/13}]]&lt;br&gt;#13
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74225" sha1="hsamy6m38vfm5u3ehzpyh6um92m89s1" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Triacontagon#Triacontagram|{15/4}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74231" sha1="tg7kqpxfavozc6bh9cfzylhc2s64gog" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74301" sha1="bh2iyex85ozyayjt1hub1nu7eioit5q" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great 5-cell digons rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |4𝝅&lt;br&gt;[[W:Pentagram|{5/2}]]&lt;br&gt;#8
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74339" sha1="ci5bxry5799e7kr5pws3uytffcnzrlq" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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        <username>Dc.samizdat</username>
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      <text bytes="74345" sha1="r6ipilhnrwjietzh4wfvbd09p32si8p" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/9}={30/12}=6{5/2}]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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        <username>Dc.samizdat</username>
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      <text bytes="74326" sha1="crg4y78ikcsv3h67yycpnzem8jg9xy3" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
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| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
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|{{radic|2.191~}}
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|- style=&quot;background: gainsboro;&quot; |
|1.345~
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|1.480~
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|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
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| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
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| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
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|{{radic|2}}
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|- style=&quot;background: seashell;&quot; |
|1.414~
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|1.414~
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|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <id>2815079</id>
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      <contributor>
        <username>Dc.samizdat</username>
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      <text bytes="74328" sha1="ord8z73fsh8tz2yh8hjxt9xwm4fcpdk" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |1200 [[5-cell#Geodesics and rotations|great digon 5-cell edges]]&lt;br&gt;(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt; in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="73904" sha1="9am9vgegbdt0pfwcdbobc2as7xrxqg7" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
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|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
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|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |400 irregular great hexagons&lt;br&gt;
(600 great rectangles)&lt;br&gt;
in 200 △ planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |Great rectangles&lt;br&gt;in &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |720 great decagons&lt;br&gt;(3600 great rectangles)&lt;br&gt;in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |200 irregular great dodecagons&lt;br&gt;(600 great rectangles)&lt;br&gt;in 200 △ planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="73832" sha1="c3pr45cfyktzjnv57ju39efcidzeysr" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |1440 [[600-cell#Decagons and pentadecagrams|great pentagons]]&lt;br&gt;(3600 great rectangles)&lt;br&gt;
in 720 &lt;big&gt;𝜙&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |4050 [[600-cell#Squares|great squares]]&lt;br&gt;
in 4050 &lt;big&gt;☐&lt;/big&gt; planes
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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        <username>Dc.samizdat</username>
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      <text bytes="73754" sha1="5n33o5n63a65baw8gispdzhv7dri0cm" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="73754" sha1="jomsz8exit8ha2oj6yjw83e9qha0c21" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;11&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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        <username>Dc.samizdat</username>
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      <text bytes="74292" sha1="egmws0yrjp76eu879uhvsoldr4odr5k" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! Rectangles
! 4-polytope
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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        <username>Dc.samizdat</username>
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      <text bytes="74496" sha1="49a4gllqmelmfbei4ayfe0q5nuqis8j" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! Polyhedron
! Rectangles
! 4-polytope
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74496" sha1="i18t0vepkru1cjkf4u2m6bmaecc1jjx" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of polyhedral section
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
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|- style=&quot;background: gainsboro;&quot; |
|1.300~
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|1.520~
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|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
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|- style=&quot;background: gainsboro;&quot; |
|1.345~
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|1.480~
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|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
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|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
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| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
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|{{radic|2}}
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|- style=&quot;background: seashell;&quot; |
|1.414~
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|1.414~
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|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>i18t0vepkru1cjkf4u2m6bmaecc1jjx</sha1>
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      <id>2815096</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
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      <text bytes="74496" sha1="dgvri53d1rzd8mqpqnvubvurkuud0z2" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedral
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74496" sha1="lbbyorxelckxj3qpcqc1ivzdnmzkzo9" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74496" sha1="s2ity6c47wk8a2q1dufr43w44vav9su" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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        <username>Dc.samizdat</username>
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      <text bytes="74496" sha1="jenj9qhovznqfxx0miillz1rgfnp7zk" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

...
{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.300~
|
|
|1.520~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.345~
|
|
|1.480~
|
|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
|
|
|{{radic|2}}
|
|
|- style=&quot;background: seashell;&quot; |
|1.414~
|
|
|1.414~
|
|
|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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      <text bytes="74492" sha1="ml7s6ebb7nagikvfrjlfe6ar8q9f7g7" xml:space="preserve">= Golden chords of the 120-cell =
{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - June 2026}}

&lt;blockquote&gt;Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.&lt;/blockquote&gt;
== Introduction ==
Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. 

Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. 

Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. 

We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope.

== Visualizing the 120-cell ==
{| class=&quot;wikitable floatright&quot; width=&quot;400&quot;
|style=&quot;vertical-align:top&quot;|[[File:120-cell.gif|200px]]&lt;br&gt;Orthographic projection of the 600-point 120-cell &lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt; performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all.
|style=&quot;vertical-align:top&quot;|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]&lt;br&gt;Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] &lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered.
|}

[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. 

The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells.

The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}}

Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all.

Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex.

== Compounds in the 120-cell ==
The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. 

The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell).

The 24-point (24-cell) also compounds by 5&lt;sup&gt;2&lt;/sup&gt; non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). 

The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5&lt;sup&gt;2&lt;/sup&gt; (75) disjoint instances of itself in the 600-point (120-cell), which contains 3&lt;sup&gt;2&lt;/sup&gt; times  5&lt;sup&gt;2&lt;/sup&gt; (225) distinct instances of the 24-point (24-cell), and  3&lt;sup&gt;3&lt;/sup&gt; times 5&lt;sup&gt;2&lt;/sup&gt; (675) distinct instances of the 8-point (16-cell). 

These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''}}

So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are &lt;math&gt;\phi^6&lt;/math&gt; as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside.

The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell.

== Thirty distinguished distances ==
The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
!rowspan=2|&lt;math&gt;c_t&lt;/math&gt;
!rowspan=2|arc
!rowspan=2|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{n}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|&lt;math&gt;\left\{p\right\}&lt;/math&gt;
!rowspan=2|&lt;small&gt;&lt;math&gt;m\left\{\frac{k}{d}\right\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Steinbach roots
!colspan=7|Chord lengths of the unit 120-cell
|-
!colspan=5|unit-radius length &lt;math&gt;c_t&lt;/math&gt;
!colspan=2|unit-edge length &lt;math&gt;c_t/c_1&lt;/math&gt;&lt;br&gt;in 120-cell of radius &lt;math&gt;c_8=\sqrt{2}\phi^2&lt;/math&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{1,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{30\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{4,1}-c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.270091&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^4}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.072949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;25.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{15\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{3-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.437016&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2} \phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.190983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{3,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;36{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{10\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3 \left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(\sqrt{5}-1\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.618034&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.381966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.28825&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;41.4{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.707107&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.61803&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{5,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;44.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \left\{\frac{15}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{2,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.756934&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{3}{2}}}{\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2 \phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.572949}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.80252&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{6,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;49.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{17}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{5-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.831254&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5}}{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.690983}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.07768&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{7,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;56.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;0.93913&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{0.881966}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.47709&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;60{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{6\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.70246&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{9,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;66.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.09132&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\chi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.04057&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{10,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;69.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1+\sqrt{5}}{2 \sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.14412&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi }{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{11,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;72{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{6}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{5\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.17557&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3-\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.38197}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \sqrt{3-\phi } \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.3525&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;75.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{24}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.22474&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.53457&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{13,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;81.1{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.30038&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{9-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.8146&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{14,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;84.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{40}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.345&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\sqrt{5} \phi }{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt[4]{5} \sqrt{\phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4.9798&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{15,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;90.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{4\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{4,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.41421&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.23607&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{16,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;95.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{29}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.4802&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.19098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.48037&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{17,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;98.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{31}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.51954&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{7+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\psi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.62605&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{18,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;104.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{8}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{4}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.58114&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{5} \sqrt{\phi ^4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{19,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;108.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{9}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{10}{3}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;c_{3,1}+c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.61803&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{1+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5.9907&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{20,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;110.2{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.64042&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13-\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.69098}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\phi ^2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.07359&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{21,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;113.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{60}{19}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.67601&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\chi }{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.20537&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{22,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;120{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{10}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{3\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.73205&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{6} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.41285&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{23,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;124.0{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{120}{41}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.7658&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4-\frac{\psi }{2 \phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.11803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\chi  \phi ^5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.53779&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{24,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;130.9{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{20}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.81907&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{11+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.30902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.73503&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{25,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;135.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{11}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{7+3 \sqrt{5}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.85123&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\phi ^2}{\sqrt{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{\phi ^4}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.42705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.8541&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{26,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;138.6{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{12}{5}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.87083&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{7}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.5}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{7} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6.92667&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{27,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;144{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{12}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{5}{2}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.90211&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\phi +2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{2+\phi }&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.61803}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{2 \phi +4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.0425&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{28,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;154.8{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{13}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.95167&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{\sqrt{13+\sqrt{5}}}{2}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.80902}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.22598&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{29,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;164.5{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{14}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{15}{7}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\phi  c_{12,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.98168&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3}{2}} \phi &lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{\frac{3 \phi ^2}{2}}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3.92705}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{3} \phi ^3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.33708&lt;/math&gt;&lt;/small&gt;
|-
|&lt;small&gt;&lt;math&gt;c_{30,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;180{}^{\circ}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{\frac{30}{15}\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\left\{2\right\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 c_{8,1}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\sqrt{4.}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2 \sqrt{2} \phi ^2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7.40492&lt;/math&gt;&lt;/small&gt;
|-
|rowspan=4 colspan=6|
|rowspan=4 colspan=4|
&lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; is the golden ratio:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi ^2-\phi -1=0&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }+1=\phi&lt;/math&gt;&lt;/small&gt;, and: &lt;small&gt;&lt;math&gt;\phi+1=\phi^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;\frac{1}{\phi }::1::\phi ::\phi ^2&lt;/math&gt;&lt;/small&gt;&lt;br&gt;
&lt;small&gt;&lt;math&gt;1/\phi&lt;/math&gt;&lt;/small&gt; and &lt;small&gt;&lt;math&gt;\phi&lt;/math&gt;&lt;/small&gt; are the golden sections of &lt;small&gt;&lt;math&gt;\sqrt{5}&lt;/math&gt;&lt;/small&gt;:&lt;br&gt;
&lt;small&gt;&lt;math&gt;\phi +\frac{1}{\phi }=\sqrt{5}&lt;/math&gt;&lt;/small&gt;
|colspan=2|&lt;small&gt;&lt;math&gt;\phi = (\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1.618034&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\chi = (3\sqrt{5} + 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = (3\sqrt{5} - 1)/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|-
|colspan=2|&lt;small&gt;&lt;math&gt;\psi = 11/\chi = 22/(3\sqrt{5} + 1)&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2.854102&lt;/math&gt;&lt;/small&gt;
|}

== Complementary chord pairs and sections ==
The list of 30 chords &lt;math&gt;c_{t}&lt;/math&gt; can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct pair of congruent [[w:120-cell#Concentric_hulls|polyhedral sections of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space &lt;math&gt;\mathbb{S}^3&lt;/math&gt;]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section of radius &lt;math&gt;c_1&lt;/math&gt; is a tetrahedron vertex figure, and the largest section of radius &lt;math&gt;c_{15}&lt;/math&gt; is a central section bisecting the 120-cell[[w:Rhombicosidodecahedron|.]] At radial distances greater than &lt;math&gt;c_{15}&lt;/math&gt; the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance &lt;math&gt;c_{29}&lt;/math&gt;. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. In Euclidean 4-dimensional space &lt;math&gt;\mathbb{R}^4&lt;/math&gt;, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]] of lateral edge length &lt;math&gt;c_{t}&lt;/math&gt;.

{| class=&quot;wikitable&quot; style=&quot;white-space:nowrap;text-align:center&quot;
! colspan=&quot;13&quot; |30 chords (15 180° pairs) make 15 kinds of section polyhedron
|-
! colspan=&quot;5&quot; |Short chord
! 4-polytope
! Polyhedron
! Rectangles
! colspan=&quot;5&quot; |Long chord
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_0&lt;/math&gt;
|0°
|
|
| rowspan=&quot;3&quot; |600 vertices&lt;br&gt;(300 axes)
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_15(2,1).svg|100px|{30/15}=15{2}]]
|180°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{30}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0}}
|
|
|{{radic|4}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0
|
|
|2
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_1&lt;/math&gt;
|15.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_polygon_30.svg|100px|{30/1}]]
| rowspan=&quot;3&quot; |[[File:120-cell_graph_H4.svg|100px|120-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_hexagons_of_the_120-cell.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,7).svg|100px|{30/14}]]
|164.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{29}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.073~}}
|
|
|{{radic|3.927~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.270~
|
|
|1.982~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_2&lt;/math&gt;
|25.2~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,1).svg|100px|{30/2}=2{15}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:25.2°_×_154.8°_chords_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-13.svg|100px|{30/13}]]
|154.8~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{28}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.191~}}
|
|
|{{radic|3.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.437~
|
|
|1.952~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_3&lt;/math&gt;
|36°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,1).svg|100px|{30/3}=3{10}]]
| rowspan=&quot;3&quot; |[[File:600-cell_graph_H4.svg|100px|600-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_decagon_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,2).svg|100px|{30/12}=6{5/2}]]
|144°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{27}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|0.382~}}
|
|
|{{radic|3.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|0.618~
|
|
|1.902~
|
|
|- style=&quot;background: gainsboro;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_4&lt;/math&gt;
|41.4~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:√0.5_×_√3.5_great_rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|138.6~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{26}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.5}}
|
|
|{{radic|3.5}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.707~
|
|
|1.871~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_5&lt;/math&gt;
|44.5~°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,2).svg|100px|{30/4}=2{15/2}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Irregular_great_dodecagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-11.svg|100px|{30/11}]]
|135.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{25}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|0.573~}}
|
|
|{{radic|3.427~}}
|
|
|- style=&quot;background: palegreen;&quot; |
|0.757~
|
|
|1.851~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_6&lt;/math&gt;
|49.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:49.1° × 130.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|130.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{24}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.691~}}
|
|
|{{radic|3.309~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.831~
|
|
|1.819~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_7&lt;/math&gt;
|56°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:56° × 124° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|124°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{23}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.882~}}
|
|
|{{radic|3.118~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|0.939~
|
|
|1.766~
|
|
|- style=&quot;background: palegreen;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_8&lt;/math&gt;
|60°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_5(6,1).svg|100px|{30/5}=5{6}]]
| rowspan=&quot;3&quot; |[[File:24-cell_t0_F4.svg|100px|24-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_hexagon.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_10(3,1).svg|100px|{30/10}=10{3}]]
|120°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{22}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1}}
|
|
|{{radic|3}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1
|
|
|1.732~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_9&lt;/math&gt;
|66.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:66.1° × 113.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|113.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{21}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.191~}}
|
|
|{{radic|2.809~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.091~
|
|
|1.676~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{10}&lt;/math&gt;
|69.8~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:69.8° × 110.2° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|110.2~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{20}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.309~}}
|
|
|{{radic|2.691~}}
|
|
|- style=&quot;background: gainsboro;&quot; |
|1.144~
|
|
|1.640~
|
|
|- style=&quot;background: yellow;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{11}&lt;/math&gt;
|72°
|
|
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_6(5,1).svg|100px|{30/6}=6{5}]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_pentagons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_3(10,3).svg|100px|{30/9}=3{10/3}]]
|108°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{19}&lt;/math&gt;
|- style=&quot;background: yellow;&quot; |
|{{radic|1.382~}}
|
|
|{{radic|2.618~}}
|
|
|- style=&quot;background: yellow;&quot; |
|1.176~
|
|
|1.618~
|
|
|- style=&quot;background: palegreen; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{12}&lt;/math&gt;
|75.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:4-simplex_t0.svg|100px|5-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_5-cell_digons_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_figure_2(15,4).svg|100px|{30/8}=2{15/4}]]
|104.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{18}&lt;/math&gt;
|- style=&quot;background: palegreen;&quot; |
|{{radic|1.5}}
|
|
|{{radic|2.5}}
|
|
|- style=&quot;background: palegreen;&quot; |
|1.224~
|
|
|1.581~
|
|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{13}&lt;/math&gt;
|81.1~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:81.1° × 98.9° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|98.9~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{17}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|1.691~}}
|
|
|{{radic|2.309~}}
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|- style=&quot;background: gainsboro;&quot; |
|1.300~
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|1.520~
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|
|- style=&quot;background: gainsboro; height:50px&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{14}&lt;/math&gt;
|84.5~°
|
|
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:84.5° × 95.5° great rectangle.png|100px]]
| rowspan=&quot;3&quot; |
|95.5~°
|
|
| rowspan=&quot;3&quot; |&lt;math&gt;c_{16}&lt;/math&gt;
|- style=&quot;background: gainsboro;&quot; |
|{{radic|0.809~}}
|
|
|{{radic|2.191~}}
|
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|- style=&quot;background: gainsboro;&quot; |
|1.345~
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|1.480~
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|
|- style=&quot;background: seashell;&quot; |
| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|90°
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|
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
| rowspan=&quot;3&quot; |[[File:4-cube_t3.svg|100px|16-cell]]
| rowspan=&quot;3&quot; |
| rowspan=&quot;3&quot; |[[File:Great_square_rectangle.png|100px]]
| rowspan=&quot;3&quot; |[[File:Regular_star_polygon_30-7.svg|100px|{30/7}]]
|90°
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| rowspan=&quot;3&quot; |&lt;math&gt;c_{15}&lt;/math&gt;
|- style=&quot;background: seashell;&quot; |
|{{radic|2}}
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|{{radic|2}}
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|- style=&quot;background: seashell;&quot; |
|1.414~
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|1.414~
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|}

== The 8-point regular polytopes ==
In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]].

A planar octagon with rigid edges of unit length has chords of length:

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613&lt;/math&gt;

The chord ratio &lt;math&gt;r_3=\sqrt{2}+1&lt;/math&gt; is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that:

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.414&lt;/math&gt;

Note that &lt;math&gt;r_3-2=1/r_3=\sqrt{2}-1&lt;/math&gt;. The procedure rotates counterclockwise over three &lt;math&gt;r_3&lt;/math&gt; chords of an {8/3} octagram. Over the first &lt;math&gt;r_3&lt;/math&gt; chord the displacement is &lt;math&gt;\sqrt{2}+r_1&lt;/math&gt;. Over the second &lt;math&gt;r_3&lt;/math&gt; chord it moves in the opposite direction a distance of &lt;math&gt;-r_1&lt;/math&gt; . Over the third &lt;math&gt;r_3&lt;/math&gt; chord it moves a distance of &lt;math&gt;-r_1&lt;/math&gt;.

If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length:

:&lt;math&gt;r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}&lt;/math&gt;

If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length:

:&lt;math&gt;r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}&lt;/math&gt;

All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius &lt;math&gt;1/\sqrt{2}&lt;/math&gt;.

[[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]]

The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron.

The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each.

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by &lt;math&gt;\sqrt{2}&lt;/math&gt; edges except opposite pairs. 

The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length:

:&lt;math&gt;r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2&lt;/math&gt;

We will need a planar octagon with rigid &lt;math&gt;r_2&lt;/math&gt; chords, rather than one with rigid &lt;math&gt;r_1&lt;/math&gt; edges. The octagon's &lt;math&gt;r_2&lt;/math&gt; chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal.

Since the edges of the 16-cell are all the same length &lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}&lt;/math&gt;, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 45° turns.

[[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell &lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt; performing a double rotation.{{Sfn|Hise|2007}}]]
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The &lt;math&gt;r_1&lt;/math&gt; chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by &lt;math&gt;r_2&lt;/math&gt; chords.

The &lt;math&gt;r_2&lt;/math&gt; chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane.

The &lt;math&gt;r_3&lt;/math&gt; chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant &lt;math&gt;r_2&lt;/math&gt; square planes at once by equal angles, moves the eight vertices along the circular helix over the &lt;math&gt;r_3&lt;/math&gt; chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not &lt;math&gt;2\pi&lt;/math&gt;, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of &lt;math&gt;r_3&lt;/math&gt; chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° &lt;math&gt;r_3&lt;/math&gt; chord makes three 45° turns. In 360° of isoclinic rotation over four &lt;math&gt;r_3&lt;/math&gt; chords, each vertex makes six 90° turns and reaches its antipodal position.

The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;6\pi&lt;/math&gt; over eight &lt;math&gt;r_3&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the four &lt;math&gt;r_2&lt;/math&gt; edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation.

Because this is the isoclinic rotation of the 16-cell in invariant planes containing its edges, we shall refer to it as the ''characteristic rotation'' of the 16-cell, and note once again that it is Fontaine and Hurley's rotation over the &lt;math&gt;r_3&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_3&lt;/math&gt;.

== Hypercubes ==
The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension &lt;math&gt;n&lt;/math&gt; is &lt;math&gt;\sqrt{n}&lt;/math&gt;, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords:
:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;
Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral.

[[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]]
The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] &lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the &lt;math&gt;\sqrt{2}&lt;/math&gt; edges of two unit-radius 16-cells, also the edges of the square central planes.

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular double helix which visits each vertex once.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing.

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}}

A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects.

== The 24-cell ==
[[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]]
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes.

The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron.

The 24-cell has the same chord set as the 4-hypercube tesseract:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}&lt;/math&gt;

[[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell &lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]]
The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a &lt;math&gt;\sqrt{3}&lt;/math&gt; chord. Each tesseract has 8 cube cells, and each cube has four &lt;math&gt;\sqrt{3}&lt;/math&gt; long diameters. The &lt;math&gt;\sqrt{3}&lt;/math&gt; chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters.

The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords:

:&lt;math&gt;r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}&lt;/math&gt;

Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: 

:&lt;math&gt;r_5-r_3+r_1+r_1-r_3=1/r_5&lt;/math&gt;

when &lt;math&gt;r_1=1&lt;/math&gt;. In the system of unit-radius coordinates &lt;math&gt;r_1=1/r_5&lt;/math&gt;. The procedure rotates counterclockwise over five &lt;math&gt;r_5&lt;/math&gt; chords of a {12/5} dodecagram.

The &lt;math&gt;r_1&lt;/math&gt; and &lt;math&gt;r_5&lt;/math&gt; chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve &lt;math&gt;\sqrt{1}&lt;/math&gt; edges each. In the skew dodecagons the chord lengths are:

:&lt;math&gt;r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 30° turns.

[[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are &lt;math&gt;\sqrt{2}&lt;/math&gt; 16-cell edges, also isocline chords in square rotations. Green chords are &lt;math&gt;\sqrt{3}&lt;/math&gt; distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]]
[[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} skew Clifford polygons of &lt;math&gt;\sqrt{2}&lt;/math&gt; great square edges in the 24-cell.]]
We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel octagon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;\sqrt{2}&lt;/math&gt; chords form a circular triple helix {24/9}=3{8/3} that visits each 24-cell vertex once.

[[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} skew Clifford polygons of &lt;math&gt;\sqrt{3}&lt;/math&gt; great hexagon diagonals in the 24-cell.]]
We can also rotate the 24-cell isoclinically by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagon rotation is a skew {12/5} dodecagram of &lt;math&gt;r_5&lt;/math&gt; chords. The rotational curve over each 120° &lt;math&gt;r_5&lt;/math&gt; chord makes five 30° turns. Two Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular double helix {24/10}=2{12/5} that visits each 24-cell vertex once. 

Fontaine and Hurley's rotation over the &lt;math&gt;r_5&lt;/math&gt; star polygon which constructs &lt;math&gt;1/r_5&lt;/math&gt; is the ''characteristic rotation of the 24-cell,'' the isoclinic rotation in invariant planes containing its edges. In the 24-cell an isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions.

The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference &lt;math&gt;10\pi&lt;/math&gt; over twelve &lt;math&gt;\sqrt{3}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane twice, over the six &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation.

== The 600-cell ==
The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron.

The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length &lt;math&gt;\phi^{-1} \approx 0.618&lt;/math&gt; instead of octahedra of edge length &lt;math&gt;\sqrt{1}&lt;/math&gt;. It encloses the &lt;math&gt;\sqrt{1}&lt;/math&gt; edges of the 24-cells, which become invisible interior chords in the 600-cell, like the &lt;math&gt;\sqrt{2}&lt;/math&gt; and &lt;math&gt;\sqrt{3}&lt;/math&gt; chords.

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of &lt;math&gt;\phi^{-1}&lt;/math&gt;, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center.

In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes.

[[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]]
The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338&lt;/math&gt;
:&lt;math&gt;r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Only the chord lengths &lt;math&gt;r_3&lt;/math&gt;, &lt;math&gt;r_5&lt;/math&gt;, &lt;math&gt;r_6&lt;/math&gt;, &lt;math&gt;\sqrt{2}&lt;/math&gt;, &lt;math&gt;r_9&lt;/math&gt;, &lt;math&gt;r_{10}&lt;/math&gt;, &lt;math&gt;r_{12}&lt;/math&gt;, &lt;math&gt;r_{15}&lt;/math&gt; occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length &lt;math&gt;r_3&lt;/math&gt;, six of which intersect in each icosahedral vertex figure. The skew {30}-gons have these chords:

:&lt;math&gt;r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618&lt;/math&gt;
:&lt;math&gt;r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}&lt;/math&gt;
:&lt;math&gt;r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_7=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176&lt;/math&gt;
:&lt;math&gt;r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}&lt;/math&gt;
:&lt;math&gt;r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618&lt;/math&gt;
:&lt;math&gt;r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}&lt;/math&gt;
:&lt;math&gt;r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902&lt;/math&gt;
:&lt;math&gt;r_{15}=2 \sin (\pi/2)=\sqrt{4}&lt;/math&gt;

Where chords are the same length, they are distinct only in the context of a rotation. The rotational curve over each &lt;math&gt;r_i&lt;/math&gt; chord makes &lt;math&gt;i&lt;/math&gt; 12° turns.

[[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell &lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt; performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]]
[[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} of &lt;math&gt;r_7&lt;/math&gt; edges.]]
We can rotate the 600-cell isoclinically in invariant planes containing 16-cell edges, by 90° in two completely orthogonal invariant square central planes, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it also intersects vertex positions of other 16-cells. Four Clifford parallel {30}-gon geodesic isoclines of circumference &lt;math&gt;6\pi&lt;/math&gt; over &lt;math&gt;r_7&lt;/math&gt; chords form a circular quadruple helix that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_7&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 16-cell'','' its isoclinic rotation in invariant planes containing 16-cell edges.

[[File:Regular_star_figure_3(10,3).svg|thumb|left|150px|{30/9}=3{10/3} of &lt;math&gt;r_9&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing 24-cell edges, by 60° in an invariant hexagon central plane and its completely orthogonal invariant central plane, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, but it does not visit other 24-cell vertex positions. Ten Clifford parallel dodecagon geodesic isoclines of circumference &lt;math&gt;10\pi&lt;/math&gt; over &lt;math&gt;\sqrt{3}&lt;/math&gt; chords form a circular helix of ten twisted strands that visits each 600-cell vertex once. [The text here cannot be quite correct; perhaps this is four {30/9}=3{10/3 as suggested by the illustration.]  Fontaine and Hurley's rotation over the &lt;math&gt;r_9&lt;/math&gt; chord is the rotation of the 600-cell by the rotation characteristic of the 24-cell'','' its isoclinic rotation in invariant planes containing 24-cell edges.

[[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} of &lt;math&gt;r_4&lt;/math&gt; edges.]]
We can also rotate the 600-cell isoclinically in invariant planes containing its own edges, by 36° in an invariant decagon central plane and its completely orthogonal invariant central plane. The Clifford polygon of the decagon rotation is a skew {15/4} pentadecagram of &lt;math&gt;r_4&lt;/math&gt; chords. Successive &lt;math&gt;r_4&lt;/math&gt; chords are edges of different 24-cells. The rotational curve over each &lt;math&gt;r_4&lt;/math&gt; chord makes five 12° turns. Eight Clifford parallel pentadecagon geodesic isoclines of circumference &lt;math&gt;5\pi&lt;/math&gt; over &lt;math&gt;\sqrt{1}&lt;/math&gt; chords form a circular helix of eight twisted strands that visits each 600-cell vertex once. Fontaine and Hurley's rotation over the &lt;math&gt;r_4&lt;/math&gt; star polygon {30/8}=2{15/4} which constructs &lt;math&gt;1/r_4&lt;/math&gt; is the characteristic rotation of the 600-cell, the isoclinic rotation in invariant planes containing its edges.   

In the 600-cell an isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon, and every great square to a Clifford parallel great square. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions.  

The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over 15 &lt;math&gt;\sqrt{1}&lt;/math&gt; chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation.

== The 5-point (5-cell) 4-simplex ==
In [[User:Dc.samizdat/Golden chords of the 120-cell#Complementary chord pairs|the table above]] Coxeter showed that the 120-cell has 30 distinct chords in 180° complementary pairs. Only 8 of those 30 chords occur in the 600-cell and the planar {30)-gon. What do the 120-cell's additional chords arise from? Originally, from the regular 5-cell, in its interaction with the other 4-polytopes that compound to make the 120-cell. Since the 120-cell and the 600-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell.

...

== Finally the 120-cell ==
The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length &lt;math&gt;c_1&lt;/math&gt;, two of which intersect in each tetrahedral vertex figure. 

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the isocclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the &lt;math&gt;\sqrt{2}&lt;/math&gt; chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.]

The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact.

== Appendix: Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|columns=7}}

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
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    <title>Media Literacy and You/The impact of the media on political economy since the time of the Pharaohs</title>
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      <text bytes="23261" sha1="lfopgccdhs6fkf2sw93cjsidz21jeve" xml:space="preserve">[[File:MaddisonDataLeaders1349-2022.svg|thumb|''Figure 1. World leaders in GDP per capita 1349-2022 (NLD, GBR, USA, SGP).&lt;ref&gt;See Bolt and van Zanden (2024) for the Maddison Data generally, van Zanden and van Leeuwen (2012) for the data on Holland 1348–1807, Smits et al. (2000) for the data on the Netherlands 1808-1913, Broadberry et al. (2015) for the data on England 1252–1700 and on Great Britain until 1870, and Sugimoto (2011) for Singapore to 2007.&lt;/ref&gt;'']]

:''I am entitled to my [[Wiktionary:cockamamie|cockamamie]] ideas, and you are entitled to yours.''&lt;ref&gt;Humor is important but must be offered in a way that does not offend others. If others are offended, they may be less interested in dialogue. The term &quot;cockamamie&quot; is used here, hoping that this style of [[w:Self-deprecation|self-deprecation]] might be more inviting for dialogue.&lt;/ref&gt; 

:This book invites you to improve your understanding of the role of the media in society and help you be more effective in talking with others and building consensus for action on the issues that most concern you. You are also invited to help improve this chapter and book. 

==Introduction== 

Acemoglu and Johnson (2023) suggest that the stability of poverty and the [[w:Malthusianism|Malthusian trap]] in hierarchical societies ''prior'' to the Industrial Revolution was enforced by &quot;societies of orders&quot; consisting of those who (1) fought, (2) prayed, and (3) worked. Those who prayed convinced those who worked to live in poverty while giving increasing shares of what they produced so those who prayed and fought could live in leisure and opulence. This seems to describe the construction of the pyramids in Egypt and the cathedrals, castles and manor homes that dot Europe today.&lt;ref&gt;Acemoglu and Johnson (2023, esp. ch. 4).&lt;/ref&gt; 

During the reign of [[w:James VI and I|King James of the King James bible]] pamphlets and newspapers began to compete with the church for helping peasants understand their role in society. That contributed to the [[w:English Civil War|English Civil War]] (1642-1651) during which James' son, [[w: Charles I of England|Charles I]], was decapitated for abuse of power. The new government allowed enough commoners to become entrepreneurs that it increased the rate of economic growth visible in Figure 1. 

The number of independent media organizations per million population continued to grow, especially in the US where they were subsidized by the US [[w:Postal Service Act|Postal Service Act]] of 1792, [[Media concentration per Columbia History Professor Richard John|until the mid nineteenth century]], when high-speed rotary presses substantially reduced the per copy cost of printing while increasing the cost of starting a newspaper.&lt;ref&gt;John (1995); John and Silberstein-Loeb (2015).&lt;/ref&gt; That contributed to the expansion of voting rights, in the US officially to all adults by 1920. During the [[w:Great Depression|Great Depression]] with over 20 percent of the US workforce unemployed, conservative arguments that blamed the poor for their poverty did not sell newspapers. That helped US President [[w:Franklin D. Roosevelt|Franklin Roosevelt]] get the political support needed for his [[w:New Deal|New Deal]] program and [[The Media, the Great Depression, and our future|wage and price controls that enabled unprecedented economic growth]] during [[w:World War II|World War II]] by dramatically limiting price gouging that had generated inflation and stifled economic growth during previous major wars in US history. 

However, the consolidation of ownership of the major media since World War II limited the increases in inequality and then reversed them starting around the time that [[w:Ronald Reagan|Ronald Reagan]] became president of the US in 1981.&lt;ref&gt;Acemoglu and Johnson (2023).&lt;/ref&gt; During the [[w:Great Recession|Great Recession]] (2007-2009) Fox featured &quot;experts&quot; who insisted that Franklin Roosevelt's New Deal made the Great Depression ''worse'', not better. That editorial distortion helped prevent the US Congress from protecting the victims of [[w:Stated income loan|liar loans]]. But the money had to be injected back into the economy, so the US Congress funded bonuses of over a million dollars each to over 5,000 finance industry leaders, some of whom should have been in prison, according to Acemoglu and Johnson (2023, ch. 3), who shared the 2024 Nobel Memorial Prize in Economics with [[w:James A. Robinson|James A. Robinson]]. A major contributor to the Great Recession has reportedly been the success of the finance industry in getting sufficient control of the major media that they have been able to dramatically reduce financial regulation codified in the [[w:Glass–Steagall legislation|Glass–Steagall legislation]] of 1933 and block other reforms like a [[w:Tobin tax|Tobin tax]] to reduce market swings from extremely short term trading, an idea for which [[w:James Tobin|James Tobin]] won the [[w:List of Nobel Memorial Prize laureates in Economic Sciences|1981 Nobel memorial prize in economics]]. 

You, dear reader, are invited to offer contrary evidence or questions regarding the evidence presented here. This chapter identifies the leading countries in [[w:List of countries by GDP (PPP) per capita|GDP per capita for each year]] in the [[w:Maddison Project|Maddison Historical Statistics Project]], which collates [[w:Gross domestic product|GDP]] per capita and population estimates for years 1 [[w:Common Era|CE]] to the present from all credible sources the project team has found. The MaddisonData package for R makes these data available as an R data object with companion functions to facilitate computing the leaders for each year with optional exclusions to facilitate identifying the technology leaders and with a function ggplotPath to make it relatively easy to plot and annotate the image as desired.&lt;ref&gt;Graves (2025).&lt;/ref&gt; The analysis then narrows the focus to apparent technology leaders since 1349, when the data identify Holland as the leader. This analysis identifies 11 different countries with the highest GDP per capita for at least one year. However, only 5 lead for more than 10 years: Holland (NLD), England / Great Britain / the United Kingdom (GBR), Australia (AUS), the United States (USA), and Singapore (SGP).&lt;ref&gt;The population of [[w:Singapore|Singapore]] is roughly 6 million. If you think that's too small to be included in this analysis, you can either ignore it or, better, repeat this analysis without it. A tutorial on how to do that using free and open-source software is available in [[Most productive countries since 1349#Appendix. Companion R Markdown vignette|an R Markdown vignette]] supplied as an appendix to the Wikiversity article on &quot;[[Most productive countries since 1349]]&quot;.&lt;/ref&gt; Australia led for only 17 of the 39 years between 1853 and 1891 as the US was overtaking the UK as the technology leader. A plot of the leaders is then made without Australia, because its inclusion would seem to add more complexity than clarity to the message. That display is a [[w:Semi-log|semilog]] plot, because it makes a constant percentage increase look like a straight line. The resulting plot of GDP per capita suggests the Industrial Revolution began in England around 1649 when King Charles I was beheaded. 

== World leaders in GDP per capita ==

Table 1 lists all the countries with the highest GDP per capita for at least one year in MaddisonData. 

{| class=&quot;wikitable&quot;
|+ ''Table 1. all the countries with the highest GDP per capita for at least one year in the MaddisonData.'' 
|-
! rowspan=2 | country !! rowspan=2 | ISO !! colspan=2 | year !! rowspan=2 | n years !! rowspan=2 | percent of years 
|-
! begin !! end  
|-
| [[w:Ancient Rome|Italy (ancient Rome)]] 
| ITA || style=&quot;text-align:right | 1 || 1501 || style=&quot;text-align:right | 3 || style=&quot;text-align:right | 0.2% 
|- 
| [[w:Iraq|Iraq]] || IRQ || style=&quot;text-align:right | 730 || 1000 || style=&quot;text-align:right | 271 || style=&quot;text-align:right | 100%
|- 
| [[w:China|China]] || CHN  || style=&quot;text-align:right | 1090 || 1150 || style=&quot;text-align:right | 61 || style=&quot;text-align:right | 100%
|- 
| [[w:England|England]] / [[w:Great Britain|Great Britain]] / [[w:United Kingdom|United Kingdom]] || GBR || style=&quot;text-align:right | 1252 || 1898 || style=&quot;text-align:right | 91 || style=&quot;text-align:right | 14% 
|-
| [[w:France|France]] || FRA || style=&quot;text-align:right | 1276 || 1374 || style=&quot;text-align:right | 19 || style=&quot;text-align:right | 19% 
|- 
| [[w:Spain|Spain]] || ESP || style=&quot;text-align:right | 1278 || 1348 || style=&quot;text-align:right | 50 || style=&quot;text-align:right | 70% 
|- 
| [[w:Sweden|Sweden]] || SWE || style=&quot;text-align:right | 1304 || 1509 || style=&quot;text-align:right | 13 || style=&quot;text-align:right | 6%
|- 
| [[w:Holland|Holland]] / [[w:Netherlands|Netherlands]] || NLD || style=&quot;text-align:right | 1349 || 1807 || style=&quot;text-align:right | 447 || style=&quot;text-align:right | 97%
|- 
| [[w:Belgium|Belgium]] || BEL || style=&quot;text-align:right | 1500 || 1500 || style=&quot;text-align:right | 1 || style=&quot;text-align:right | 100%
|- 
| [[w:Australia|Australia]] || AUS || style=&quot;text-align:right | 1853 || 1891 || style=&quot;text-align:right | 17 || style=&quot;text-align:right | 44%
|- 
| [[w:New Zealand|New Zealand]] || NZL || style=&quot;text-align:right | 1873 || 1874 || style=&quot;text-align:right | 2 || style=&quot;text-align:right | 100%
|-
| [[w:United States|United States]] || USA || style=&quot;text-align:right | 1882 || 1990 || style=&quot;text-align:right | 58 || style=&quot;text-align:right | 53%
|- 
| [[w:Switzerland|Switzerland]] || CHE || style=&quot;text-align:right | 1931 || 1934 || style=&quot;text-align:right | 4 || style=&quot;text-align:right | 100%
|- 
| [[w:Qatar|Qatar]] || QAT || style=&quot;text-align:right | 1950 || 2022 || style=&quot;text-align:right | 45 || style=&quot;text-align:right | 62%
|- 
| [[w:Kuwait|Kuwait]] || KWT || style=&quot;text-align:right | 1953 || 1957 || style=&quot;text-align:right | 5 || style=&quot;text-align:right | 100% 
|-
|[[w:United Arab Emirates|United Arab Emirates]] || ARE || style=&quot;text-align:right | 1965 || 1984 || style=&quot;text-align:right | 5 || style=&quot;text-align:right | 25%
|- 
| [[w:Luxembourg|Luxembourg]] || LUX || style=&quot;text-align:right | 1991 || 1995 || style=&quot;text-align:right | 5 || style=&quot;text-align:right | 100%
|- 
| [[w:Norway|Norway]] || NOR || style=&quot;text-align:right | 1996 || 2002 || style=&quot;text-align:right | 7 || style=&quot;text-align:right | 100% 
|} 

For simplicity, we focus on the data since 1349, when Holland had the highest GDP per capita, omitting Qatar (QAT), Kuwait (KWT), United Arab Emirates (ARE), and Norway (NOR), whose wealth may be more due to petroleum than to broad technology leadership. We also delete Luxenbourg (LUX), whose population is under a million and therefore may be too small to use for general conclusions about technology leadership. Table 2 is similar to Table 1 with these adjustments. 

{| class=&quot;wikitable&quot;
|+ ''Table 2. Highest GDP per capita among broad-based economies since 1349.'' 
|-
! rowspan=2 | country !! rowspan=2 | ISO !! colspan=2 | year !! rowspan=2 | n years !! rowspan=2 | percent of years 
|-
! begin !! end  
|- 
| [[w:Holland|Holland]] / [[w:Netherlands|Netherlands]] || NLD || 1349 || 1807 || style=&quot;text-align:right | 447 || style=&quot;text-align:right | 97%
|- 
| [[w:France|France]] || FRA || 1357 || 1374 || style=&quot;text-align:right | 7 || style=&quot;text-align:right | 39% 
|- 
| [[w:Italy|Italy]] || ITA || 1451 || 1501 || style=&quot;text-align:right | 2 || style=&quot;text-align:right | 4%
|- 
| [[w:Sweden|Sweden]] || SWE || 1468 || 1509 || style=&quot;text-align:right | 2 || style=&quot;text-align:right | 5%
|- 
| [[w:Belgium|Belgium]] || BEL || 1500 || 1500 || style=&quot;text-align:right | 1 || style=&quot;text-align:right | 100%
|- 
| [[w:England|England]] / [[w:Great Britain|Great Britain]] / [[w:United Kingdom|United Kingdom]] || GBR || 1808 || 1898 || style=&quot;text-align:right | 67 || style=&quot;text-align:right | 74% 
|- 
| [[w:Australia|Australia]] || AUS || 1853 || 1891 || style=&quot;text-align:right | 17 || style=&quot;text-align:right | 44%
|- 
| [[w:New Zealand|New Zealand]] || NZL || 1873 || 1874 || style=&quot;text-align:right | 2 || style=&quot;text-align:right | 100%
|- 
| [[w:United States|United States]] || USA || 1882 || 2005 || style=&quot;text-align:right | 107 || style=&quot;text-align:right | 86% 
|- 
| [[w:Switzerland|Switzerland]] || CHE || 1931 || 2009 || style=&quot;text-align:right | 9 || style=&quot;text-align:right | 11%
|- 
| [[w:Singapore|Singapore]] || SGP || 2010 || 2022 || style=&quot;text-align:right | 13 || style=&quot;text-align:right | 100%
|}

Singapore (SGP) has replaced Norway as the current leader, according to the Maddison project data. The Wikipedia article on &quot;[[w:List of countries by GDP (PPP) per capita|List of countries by GDP (PPP) per capita]]&quot;&lt;ref&gt;accessed 2025-01-01&lt;/ref&gt; notes that data from the US [[w:Central Intelligence Agency|Central Intelligence Agency]] report GDP per capita numbers for [[w:Monaco| Monaco]] (MCO) and [[w:Liechtenstein|Liechtenstein]] (LIE) higher than for Singapore and Norway. However, they are tiny countries with populations roughly 40,000 each and are not included in MaddisonData. 

[[w:Holland|Holland]] (NLD) was the leader for 97 percent of the years between 1349 and 1807, according to MaddisonData. Then between 1807 and 1808, GDP per capita for NLD fell by 32 percent -- almost a third. That change can be attributed at least in part to a change in the definition of &quot;NLD&quot;: Up to 1807, NLD represented Holland, per van Zanden and van Leeuwen (2012). Beginning in 1808, the data are for the [[w:Netherlands|Netherlands]], per Smits et al. (2000), of which Holland is only part. Those years were also during the [[w:Napoleonic Wars|Napoleonic Wars]], and the Netherlands were part of France for part of that period. To understand this drop better, we would need to consult experts on that history.   

After that change, Holland / the Netherlands was replaced as the leader in GDP per capita by England / Great Britain / the United Kingdom (GBR), which led for 74 percent of the 91 years between 1808 and 1898. Then the US led for 84 percent of the years between 1882 and 1990 with Australia (AUS), New Zealand (NZL) and Switzerland (CHE) leading for the remaining 16 percent of those years. Luxembourg (LUX) led between 1991 and 2008, then Switzerland (CHE) led for 2009, then Singapore (SGP) between 2010 and 2022. 

The next section discusses a plot of the data for NLD, GBR, USA, and SGP. Others countries are omitted, because their leadership was so short, according to these data, that including them might add more complexity than information and make it harder to understand the big picture.

== Plot broad-based leaders == 

Figure 1 is a [[w:Semi-log|semilog]] plot of GDP per capita for NLD, GBR, USA, and SGP between 1349 and 2022. A semilog plot like this makes a constant percentage increase appear as a straight line. Annotations document some of the potentially most important events during this period: 
* The orange line represents Holland through 1807 and the Netherlands starting in 1808. 
* The English Civil War (1642-1652), during which King Charles I was decapitated (1649). 
* The War of 1912 (1812-1815). 
* The American Civil War (1861-1865). 
* [[w:World War I|WW1]] (1914-1918). 
* The presidency of Herbert Hoover (1929-1933). 
* The presidency of Franklin Roosevelt (1933-1945). 
* [[w:World War II|WW2]] (1939-1945). 
* The presidency of Ronald Reagan (1981-1989). 
* The first presidency of Donald Trump (2017-2021). 
* The presidency of Joe Biden (2021-2025). 

The orange NLD line includes a drop of 32 percent between 1807 and 1808 as the data changed from representing only Holland to representing the Netherlands, as mentioned above. 

A key feature of a semilog plot is that a constant percentage increase appears as a straight line with the slope being proportional to the rate of growth. A fairly obvious feature of Figure 1 is that GDP per capita started increasing for England very  close to 1649, which King Charles I lost his head. England combined with Scotland to become Great Britain by the [[w:Acts of Union 1707|Acts of Union of 1707]] during the reign of [[w:Anne, Queen of Great Britain|Queen Ann]], which was accompanied by economic turbulence visible in Figure 1. After she died, the economy began growing again but at a slower rate. Great Britain merged with Ireland by the [[w:Acts of Union 1800|Acts of Union of 1800]] to become the United Kingdom (UK) during the reign of George III. The creation of the UK was quickly followed by the [[w:Napoleonic Wars|Napoleonic Wars]] (1803-1815), which included the [[w:War of 1812|War of 1812]], which is marked on this plot. Those wars were followed by a brief decline in the UK GDP per capita, but it quickly started growing again at a faster rate. Both World Wars had negative impacts on the UK economy, visible in Figure 1. 

GDP per capita for the US started well below that of the UK, to the extent that the Maddison data are accurate, but grew faster and overtook the UK between 1882 and 1898, according to Table 2 above. The most spectacular features in Figure 1 are the unprecedented decline of the US economy during the administration of Herbert Hoover followed an even more unprecedented increase during the administration of FDR. 

The rate of growth in GDP per capita in the US is visibly slowing before Singapore takes the lead at the beginning of the Great Recession. [[w:Thomas Piketty|Thomas Piketty]], the world's leading expert on inequality, has attributed that slowing of the US economy to the increase in inequality since Reagan became US president in 1981, documented in Figures 6 and 7 of the next chapter of this book on, &quot;[[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future|Fox, the Great Depression, the Great Recession, and our future]]&quot;. That chapter includes a section on the &quot;[[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future#Role of the media|Role of the media]]&quot;, which cites research suggesting that both the increase in inequality and the slowing of the rate of economic growth can be attributed to the increased concentration of ownership of the major media including for-profit social media, which make money increasing political polarization and violence.

== Caveat ==

Lindert and Williamson insist that Maddison's data are deficient, at least regarding the relative position of the 13 colonies that became the US: {{quote| 

American world leadership in income per person has waxed and waned for centuries.
Before the twentieth century, the period in which Americans most clearly led Britain and all of western Europe in purchasing power per capita was during colonial times—that is, when North Americans were still British. They were already ahead by the late seventeenth century. America lost that lead in the Revolutionary War and the Articles of Confederation years, gained it back by 1860, lost most of it again in the Civil War decade, gained it back once more by 1900, and briefly lost it again in the Great Depression of the 1930s. ''Angus Maddison’s claim that American income per capita did not catch up to that of Britain until the start of the twentieth century seems to be off the mark by at least two centuries.''

Over the whole span of over 360 years since the mid-seventeenth century, America’s income advantage over Britain has not increased and may have decreased slightly. The only historical moment in which the United States soared far ahead of the rest of the world in average income came at the end of World War II. Since then, western Europe and Japan have been growing faster than the United States in terms of incomes per person. (emphasis added.)&lt;ref&gt;Lindert and Williamson (2016, pp. 8-9).&lt;/ref&gt;}}

This challenge to the numbers in the current analysis is vital for understanding the impact of armed conflict on the economy but may be irrelevant to the main point of this book, that media play a major role in helping humans understand what they should do to advance their interests.

== Exercise ==

Share with others your thoughts on the issues raised in this discussion and summarize those discussions on the &quot;Discuss&quot; page associated with this chapter. Focus especially on how you managed your emotions and your relationships with the humans with whom you spoke as well as the strengths and weaknesses in the content of this chapter, challenges that should be addressed, and suggestions for improvement. 

== See also ==

* [[Most productive countries since 1349]]
* [[The Media, the Great Depression, and our future]]

== Notes ==
{{reflist}}

== Bibliography ==

* [[d: Q125292212|Daron Acemoğlu and Simon Johnson (2023) ''Power and Progress: Our Thousand-Year Struggle Over Technology and Prosperity'' (PublicAffairs)]]. 

* [[d:Q126723821|Jutta Bolt and Jan Luiten van Zanden (2024) &quot;Maddison style estimates of the evolution of the world economy: A new 2023 update&quot;, Journal of Economic Surveys, 1-41]]. 

* [[d:Q57945943|S. N. Broadberry, B. Campbell, A. Klein, M. Overton and B. van Leeuwen (2015) ''British Economic Growth 1270-1870'' (Cambridge University Press)]].

* [[d:Q137660377|Susan B. Carter, S. S. Gartner, M. R. Haineset (2006) ''Historical Statistics of the United States: Earliest Time to the Present'' (Cambridge University Press)]]. 

* [[d:Q137660514|Spencer Graves (2025-11-25) &quot;MaddisonData: Maddison Project Data&quot; software available from the Comprehensive R Archive Network (CRAN) and GitHub]]. 

* [[d:Q54641943|Richard R. John (1995) ''Spreading the News: The American Postal System from Franklin to Morse'' (Harvard University Press)]]. 

* [[d:Q131468166|Richard R. John; Jonathan Silberstein-Loeb, eds. (2015) ''Making News: The Political Economy of Journalism in Britain and America from the Glorious Revolution to the Internet'' (Oxford University Press)]].

* [[d:Q137669937|John J. McCusker (2006) &quot;Colonial Statistics&quot;, Carter et al. (2006, V-671)]].  

* [[d:Q135527962|Reece Peck (2016) &quot;Usurping the usable past: How Fox News remembered the Great Depression during the Great Recession&quot;, ''Journalism'', 18(6)]].

* [[d:Q55878109|W. Scheidel and S. J. Friesen (2009) &quot;The size of the economy and the distribution of income in the Roman Empire&quot;, ''Journal of Roman Studies'', 99, pp. 61–91]]. 

* [[d:Q137669960|J.P. Smits, E. Horlings and J.L. van Zanden (2000) Dutch GDP and its Components 1800-1913 (Groningen Growth and Development Centre)]]. 

* [[d:Q137669987|I. Sugimoto (2011) ''Economic growth of Singapore in the twentieth century: historical GDP estimates and empirical investigations'' ( World Scientific Publishing)

* [[d: Q137670038|R. Sutch (2006). National Income and Product. Carter et al. (2006, III-23-25)]].

* [[d: Q137670058|J. L. van Zanden  and B. van Leeuwen (2012), ‘Persistent but not consistent: the growth of national income in Holland 1347–1807’, Explorations in Economic History, 49, pp. 119–30]].

[[Category:Media literacy]]
[[Category:Communication]]
[[Category:Political science]]
[[Category:Law]]
[[Category:Education]]
[[Category:Economics]]
[[Category:Media Literacy and You]]
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Hello and welcome to my user page. I'm AUBSTRAWBS, I like history, art, aerospace and physics and especially any subject that somehow manages to combine them all. Also i really like anime which will be made obvious later on.

== About me ==
I love divulging my personal info online lol.

=== Stuff ===

=== Sports ===

=== Projects ===

=== Online presence ===

== People I think are cool ==
(Disclaimer I have not fully researched these people it is possible they have done bad things that I don't agree with. But with everything I have read about them it seems they be good.)

=== Marc Isambar(d/t) Brunel ===
I learnt about this man when i was looking for a subject for a presentation in school. He did many super cool things such as automating the production of pulley blocks, making super beautiful designs for the US Capitol building and building the first tunnel under the river Thames.  Also he was the father of Isambard Kingdom Brunel and he has been unfortunately hella overshadowed by his son. [[File:CapitolBuildingMarcBrunel.jpg|thumb|center|250px|Marc Isambard Brunel's design of the capitol building]]

== Jokes ==

=== Stutter ===
Wanna hear a joke, a a a a man walks into a bar. The Bar tender: couldn't have said it better myself.       

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Wanna hear another joke, a horse walks into a bar, the bartender asks do you wanna hear a joke. The horse says yes, so the bartender pulls out the 2 of diamonds 3 of diamonds 4 of diamonds 5 of diamonds 6 of diamonds 7 of diamonds 8 of diamonds 9 of diamonds 10 of diamonds jack of diamonds queen of diamonds king of diamonds and the ace of diamonds then the 2 of clubs 3 of clubs 4 of clubs 5 of clubs 6 of clubs 7 of clubs 8 of clubs 9 of clubs 10 of clubs jack of clubs queen of clubs king of clubs and the ace of clubs then the 2 of spades 3 of spades 4 of spades 5 of spades 6 of spades 7 of spades 8 of spades 9 of spades 10 of spades jack of spades queen of spades king of spades and the ace of spades then the 2 of hearts 3 of hearts 4 of hearts 5 of hearts 6 of hearts 7 of hearts 8 of hearts 9 of hearts 10 of hearts jack of hearts queen of hearts king of hearts but I'm missing the ace of hearts can I have your heart to complete my set. The horse says neigh and walks away.               [[File:Cat with open mouth.jpg|thumb|User's cat for user page]]

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Hello and welcome to my user page. I'm AUBSTRAWBS, I like history, art, aerospace and physics and especially any subject that somehow manages to combine them all. Also i really like anime which will be made obvious later on.

== About me ==
I love divulging my personal info online lol.

=== Stuff ===

=== Sports ===

=== Projects ===

=== Online presence ===

== People I think are cool ==
(Disclaimer I have not fully researched these people it is possible they have done bad things that I don't agree with. But with everything I have read about them it seems they be good.)

=== Marc Isambar(d/t) Brunel ===
I learnt about this man when i was looking for a subject for a presentation in school. He did many super cool things such as automating the production of pulley blocks, making super beautiful designs for the US Capitol building and building the first tunnel under the river Thames.  Also he was the father of Isambard Kingdom Brunel and he has been unfortunately hella overshadowed by his son. [[File:CapitolBuildingMarcBrunel.jpg|thumb|center|250px|Marc Isambard Brunel's design of the capitol building]]

== Jokes ==

=== Stutter ===
Wanna hear a joke, a a a a man walks into a bar. The Bar tender: couldn't have said it better myself.       

=== Cards ===
Wanna hear another joke, a horse walks into a bar, the bartender asks do you wanna hear a joke. The horse says yes, so the bartender pulls out the 2 of diamonds 3 of diamonds 4 of diamonds 5 of diamonds 6 of diamonds 7 of diamonds 8 of diamonds 9 of diamonds 10 of diamonds jack of diamonds queen of diamonds king of diamonds and the ace of diamonds then the 2 of clubs 3 of clubs 4 of clubs 5 of clubs 6 of clubs 7 of clubs 8 of clubs 9 of clubs 10 of clubs jack of clubs queen of clubs king of clubs and the ace of clubs then the 2 of spades 3 of spades 4 of spades 5 of spades 6 of spades 7 of spades 8 of spades 9 of spades 10 of spades jack of spades queen of spades king of spades and the ace of spades then the 2 of hearts 3 of hearts 4 of hearts 5 of hearts 6 of hearts 7 of hearts 8 of hearts 9 of hearts 10 of hearts jack of hearts queen of hearts king of hearts but I'm missing the ace of hearts can I have your heart to complete my set. The horse says neigh and walks away.               [[File:Cat with open mouth.jpg|thumb|User's cat for user page]]

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Hello and welcome to my user page. I'm AUBSTRAWBS, I like history, art, aerospace and physics and especially any subject that somehow manages to combine them all. Also i really like anime which will be made obvious later on.

== About me ==
I love divulging my personal info online lol.

=== Stuff ===

=== Sports ===

=== Projects ===

=== Online presence ===

== People I think are cool ==
(Disclaimer I have not fully researched these people it is possible they have done bad things that I don't agree with. But with everything I have read about them it seems they be good.)

=== Marc Isambar(d/t) Brunel ===
I learnt about this man when i was looking for a subject for a presentation in school. He did many super cool things such as automating the production of pulley blocks, making super beautiful designs for the US Capitol building and building the first tunnel under the river Thames.  Also he was the father of Isambard Kingdom Brunel and he has been unfortunately hella overshadowed by his son. [[File:CapitolBuildingMarcBrunel.jpg|thumb|center|250px|Marc Isambard Brunel's design of the capitol building]]

== Jokes ==

=== Cards ===
Wanna hear another joke, a horse walks into a bar, the bartender asks do you wanna hear a joke. The horse says yes, so the bartender pulls out the 2 of diamonds 3 of diamonds 4 of diamonds 5 of diamonds 6 of diamonds 7 of diamonds 8 of diamonds 9 of diamonds 10 of diamonds jack of diamonds queen of diamonds king of diamonds and the ace of diamonds then the 2 of clubs 3 of clubs 4 of clubs 5 of clubs 6 of clubs 7 of clubs 8 of clubs 9 of clubs 10 of clubs jack of clubs queen of clubs king of clubs and the ace of clubs then the 2 of spades 3 of spades 4 of spades 5 of spades 6 of spades 7 of spades 8 of spades 9 of spades 10 of spades jack of spades queen of spades king of spades and the ace of spades then the 2 of hearts 3 of hearts 4 of hearts 5 of hearts 6 of hearts 7 of hearts 8 of hearts 9 of hearts 10 of hearts jack of hearts queen of hearts king of hearts but I'm missing the ace of hearts can I have your heart to complete my set. The horse says neigh and walks away.               [[File:Cat with open mouth.jpg|thumb|User's cat for user page]]

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Hello and welcome to my user page. I'm AUBSTRAWBS, I like history, art, aerospace and physics and especially any subject that somehow manages to combine them all. Also i really like anime which will be made obvious later on.

== About me ==
I love divulging my personal info online lol.

=== Stuff ===

=== Sports ===

=== Projects ===

=== Online presence ===

== People I think are cool ==
(Disclaimer I have not fully researched these people it is possible they have done bad things that I don't agree with. But with everything I have read about them it seems they be good.)

=== Marc Isambar(d/t) Brunel ===
I learnt about this man when i was looking for a subject for a presentation in school. He did many super cool things such as automating the production of pulley blocks, making super beautiful designs for the US Capitol building and building the first tunnel under the river Thames.  Also he was the father of Isambard Kingdom Brunel and he has been unfortunately overshadowed by his son. [[File:CapitolBuildingMarcBrunel.jpg|thumb|center|250px|Marc Isambard Brunel's design of the capitol building]]

== Jokes ==

=== Cards ===
Wanna hear another joke, a horse walks into a bar, the bartender asks do you wanna hear a joke. The horse says yes, so the bartender pulls out the 2 of diamonds 3 of diamonds 4 of diamonds 5 of diamonds 6 of diamonds 7 of diamonds 8 of diamonds 9 of diamonds 10 of diamonds jack of diamonds queen of diamonds king of diamonds and the ace of diamonds then the 2 of clubs 3 of clubs 4 of clubs 5 of clubs 6 of clubs 7 of clubs 8 of clubs 9 of clubs 10 of clubs jack of clubs queen of clubs king of clubs and the ace of clubs then the 2 of spades 3 of spades 4 of spades 5 of spades 6 of spades 7 of spades 8 of spades 9 of spades 10 of spades jack of spades queen of spades king of spades and the ace of spades then the 2 of hearts 3 of hearts 4 of hearts 5 of hearts 6 of hearts 7 of hearts 8 of hearts 9 of hearts 10 of hearts jack of hearts queen of hearts king of hearts but I'm missing the ace of hearts can I have your heart to complete my set. The horse says neigh and walks away.               [[File:Cat with open mouth.jpg|thumb|User's cat for user page]]

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Hello and welcome to my user page. I'm AUBSTRAWBS, I like history, art, aerospace and physics and especially any subject that somehow manages to combine them all. Also i really like anime which will be made obvious later on.

== About me ==
I love divulging my personal info online lol.
One of my main goals on wikiversity is to bring over as many userboxes as possible from wikipedia en. However i won't bring over porn, racist, or offensive ones just cuz i dont want to.

=== Stuff ===

=== Sports ===

=== Projects ===

=== Online presence ===

== People I think are cool ==
(Disclaimer I have not fully researched these people it is possible they have done bad things that I don't agree with. But with everything I have read about them it seems they be good.)

=== Marc Isambar(d/t) Brunel ===
I learnt about this man when i was looking for a subject for a presentation in school. He did many super cool things such as automating the production of pulley blocks, making super beautiful designs for the US Capitol building and building the first tunnel under the river Thames.  Also he was the father of Isambard Kingdom Brunel and he has been unfortunately overshadowed by his son. [[File:CapitolBuildingMarcBrunel.jpg|thumb|center|250px|Marc Isambard Brunel's design of the capitol building]]

== Jokes ==

=== Cards ===
Wanna hear another joke, a horse walks into a bar, the bartender asks do you wanna hear a joke. The horse says yes, so the bartender pulls out the 2 of diamonds 3 of diamonds 4 of diamonds 5 of diamonds 6 of diamonds 7 of diamonds 8 of diamonds 9 of diamonds 10 of diamonds jack of diamonds queen of diamonds king of diamonds and the ace of diamonds then the 2 of clubs 3 of clubs 4 of clubs 5 of clubs 6 of clubs 7 of clubs 8 of clubs 9 of clubs 10 of clubs jack of clubs queen of clubs king of clubs and the ace of clubs then the 2 of spades 3 of spades 4 of spades 5 of spades 6 of spades 7 of spades 8 of spades 9 of spades 10 of spades jack of spades queen of spades king of spades and the ace of spades then the 2 of hearts 3 of hearts 4 of hearts 5 of hearts 6 of hearts 7 of hearts 8 of hearts 9 of hearts 10 of hearts jack of hearts queen of hearts king of hearts but I'm missing the ace of hearts can I have your heart to complete my set. The horse says neigh and walks away.               [[File:Cat with open mouth.jpg|thumb|User's cat for user page]]

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== Examples ==

# Let's consider a first-order nonlinear equation that represents an organism's population:
#:
#: &lt;math&gt;\frac {dP}{dt} = 0.6P(1 - \frac {P}{540})&lt;/math&gt;
#:
#: a) What are the equilibrium population states?
#: b) At what populations is the population increasing?
#: c) At what populations is the population decreasing?
#Consider another nonlinear equation representing population:
#:&lt;math&gt;\frac {dP}{dt} = 0.8P(1 - \frac {P}{80})(\frac {P}{30}-1)P&lt;/math&gt;
#:
#: a) What are the equilibrium population states?
#: b) At what populations is the population increasing?
#: c) At what populations is the population decreasing?

{{BookCat}}</text>
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      <contributor>
        <username>JackBot</username>
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      <minor />
      <comment>Bot: Fixing double redirect from [[Template:Gate-anime]] to [[Template:User gate anime]]</comment>
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  <page>
    <title>User talk:Jessephu</title>
    <ns>3</ns>
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      <contributor>
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        <id>276019</id>
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      <text bytes="2311" sha1="2ifub6cxmr3g2bat6zdmeo4p9pkhaqw" xml:space="preserve">==Welcome==
{{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]],  Jessephu!'''|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Atcovi|me personally]] if you would like some [[Help:Contents|help]].

Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple.

We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies.

To find your way around, check out:
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Wikiversity:Introduction|Introduction to Wikiversity]]
* [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]]
* [[Wikiversity:Browse|Browse]] or visit an educational level portal:&lt;br&gt;[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]]
* [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;

* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]]
* Learn [[Help:How to write an educational resource|how to write an educational resource]]
* Find out about [[Wikiversity:Research|research]] activities
* Give [[Wikiversity:Feedback|feedback]] about your observations
* Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]].

See you around Wikiversity! --—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:34, 10 June 2026 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}</text>
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|Description=Carry Lookahead Adders 2A traditional (20260610 - 20260609)
|Source={{own|Young1lim}}
|Date=2026-06-10
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|Source={{own|Young1lim}}
|Date=2026-06-10
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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|Description=C04.SA0: Address and Dereference Operators (20260610 - 20260609)
|Source={{own|Young1lim}}
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|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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|Description=LCal.9A: Recursion (20260608 - 20260603)
|Source={{own|Young1lim}}
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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|Description=LCal.9A: Recursion (20260609 - 20260608)
|Source={{own|Young1lim}}
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|Author=Young W. Lim
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}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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== Licensing ==
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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| border-c = #aaf0a1
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| info-c   = #aaf0a1
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      <contributor>
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      <contributor>
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        <id>3060598</id>
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      <comment>bring over all anime related user boxes</comment>
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      <comment>removed non existant link</comment>
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}}&lt;noinclude&gt;&lt;br style=&quot;clear: both;&quot;&gt;</text>
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| info-c   = #F0F8FF
| info-s   = 8
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}}&lt;includeonly&gt;{{
&lt;/includeonly&gt;&lt;noinclude&gt;&lt;br style=&quot;clear: both;&quot;&gt;</text>
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    <revision>
      <id>2815110</id>
      <timestamp>2026-06-10T20:39:14Z</timestamp>
      <contributor>
        <username>AUBSTRAWBS</username>
        <id>3060598</id>
      </contributor>
      <comment>Created page with &quot;{{userbox | border-c = #77E0E8 | id       = [[:Category:Wikipedians interested in anime and manga|o_O]] | id-c     = #77E0E8 | id-s     = 14 | info     = This user is a moderate fan of '''[[anime]]'''. | info-c   = #D0F8FF | info-s   = 8 | info-lh  = 1.25em }}&lt;includeonly&gt;&quot;</comment>
      <origin>2815110</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="273" sha1="h8jc4h18tehn1o3b9jp130vxrbol3pb" xml:space="preserve">{{userbox
| border-c = #77E0E8
| id       = [[:Category:Wikipedians interested in anime and manga|o_O]]
| id-c     = #77E0E8
| id-s     = 14
| info     = This user is a moderate fan of '''[[anime]]'''.
| info-c   = #D0F8FF
| info-s   = 8
| info-lh  = 1.25em
}}&lt;includeonly&gt;</text>
      <sha1>h8jc4h18tehn1o3b9jp130vxrbol3pb</sha1>
    </revision>
  </page>
  <page>
    <title>Template:User anime User anime-3</title>
    <ns>10</ns>
    <id>330124</id>
    <revision>
      <id>2815111</id>
      <timestamp>2026-06-10T20:41:03Z</timestamp>
      <contributor>
        <username>AUBSTRAWBS</username>
        <id>3060598</id>
      </contributor>
      <comment>Created page with &quot;{{userbox | border-c = #99B3FF | id       = [[:Category:Wikipedians interested in anime and manga|~_~]] | id-c     = #99B3FF | id-s     = 14 | info     = This user is a definite fan of '''[[anime]]'''. | info-c   = #E0E8FF | info-s   = 8 | info-lh  = 1.25em }}&lt;includeonly&gt;&quot;</comment>
      <origin>2815111</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="273" sha1="0lsqu7e62h7n66s4zk7e6z6etirgi2r" xml:space="preserve">{{userbox
| border-c = #99B3FF
| id       = [[:Category:Wikipedians interested in anime and manga|~_~]]
| id-c     = #99B3FF
| id-s     = 14
| info     = This user is a definite fan of '''[[anime]]'''.
| info-c   = #E0E8FF
| info-s   = 8
| info-lh  = 1.25em
}}&lt;includeonly&gt;</text>
      <sha1>0lsqu7e62h7n66s4zk7e6z6etirgi2r</sha1>
    </revision>
  </page>
  <page>
    <title>Template:User anime User anime-4</title>
    <ns>10</ns>
    <id>330125</id>
    <revision>
      <id>2815112</id>
      <timestamp>2026-06-10T20:42:33Z</timestamp>
      <contributor>
        <username>AUBSTRAWBS</username>
        <id>3060598</id>
      </contributor>
      <comment>Created page with &quot;{{userbox | border-c = #CCCC00 | id       = [[:Category:Wikipedians interested in anime and manga|*_*]] | id-c     = #FFFF00 | id-s     = 14 | info     = This user loves [[anime]]'''. | info-c   = #FFFF99 | info-s   = 8 | info-lh  = 1.25em }}&lt;includeonly&gt;&quot;</comment>
      <origin>2815112</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="255" sha1="fyydj66npbqphrz6iett15hbrlnc9lw" xml:space="preserve">{{userbox
| border-c = #CCCC00
| id       = [[:Category:Wikipedians interested in anime and manga|*_*]]
| id-c     = #FFFF00
| id-s     = 14
| info     = This user loves [[anime]]'''.
| info-c   = #FFFF99
| info-s   = 8
| info-lh  = 1.25em
}}&lt;includeonly&gt;</text>
      <sha1>fyydj66npbqphrz6iett15hbrlnc9lw</sha1>
    </revision>
  </page>
  <page>
    <title>Template:User/anime/User anime-N</title>
    <ns>10</ns>
    <id>330126</id>
    <revision>
      <id>2815113</id>
      <timestamp>2026-06-10T20:47:04Z</timestamp>
      <contributor>
        <username>AUBSTRAWBS</username>
        <id>3060598</id>
      </contributor>
      <comment>Created page with &quot;{{userbox-level | level    =  | id       = [[:Category:Wikipedians interested in anime and manga|アニメ]] | id-s     = 10 | info     = This user is an '''[[anime]] otaku. }}&lt;includeonly&gt;&quot;</comment>
      <origin>2815113</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="189" sha1="t07j4qasmn94q828qemky8a42vst3c1" xml:space="preserve">{{userbox-level
| level    = 
| id       = [[:Category:Wikipedians interested in anime and manga|アニメ]]
| id-s     = 10
| info     = This user is an '''[[anime]] otaku.
}}&lt;includeonly&gt;</text>
      <sha1>t07j4qasmn94q828qemky8a42vst3c1</sha1>
    </revision>
  </page>
  <page>
    <title>Template:User/anime/Better actors</title>
    <ns>10</ns>
    <id>330127</id>
    <revision>
      <id>2815114</id>
      <timestamp>2026-06-10T20:49:49Z</timestamp>
      <contributor>
        <username>AUBSTRAWBS</username>
        <id>3060598</id>
      </contributor>
      <comment>personally i think most dubs are pretty good</comment>
      <origin>2815114</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="409" sha1="jlic5we9p5e8yqg2gx2c5bsl1psuoe6" xml:space="preserve">&lt;div style=&quot;float:left;border:solid black 1px;margin:1px&quot;&gt;
{| cellspacing=&quot;0&quot; style=&quot;width:238px;background:#AAA&quot;
| style=&quot;width:45px;height:45px;background:red;color:white;text-align:center;font-size:19pt&quot; | '''-_-¿'''
| style=&quot;font-size:8pt;padding:4pt;line-height:1.25em&quot; | &lt;span style=&quot;color:white&quot;&gt;This user wishes that [[Dubbing (filmaking)|dubbed]] [[anime]] could have better actors.&lt;/span&gt;
|}
&lt;/div&gt;</text>
      <sha1>jlic5we9p5e8yqg2gx2c5bsl1psuoe6</sha1>
    </revision>
  </page>
  <page>
    <title>Template:User/manga/User manga</title>
    <ns>10</ns>
    <id>330128</id>
    <revision>
      <id>2815115</id>
      <timestamp>2026-06-10T20:55:13Z</timestamp>
      <contributor>
        <username>AUBSTRAWBS</username>
        <id>3060598</id>
      </contributor>
      <comment>personally i like oshi no ko</comment>
      <origin>2815115</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="319" sha1="5v0kyowghfdih2td4xy75styqsipdx7" xml:space="preserve">{{userbox
| border-c = #6EF7A7
| id       = [[:Category:Wikipedians interested in anime and manga|^_^]]
| id-c     = #6EF7A7
| id-s     = 14
| info     = This user reads manga.
| info-c   = #C5FCDC
| info-s   = 8
| info-lh  = 1.25em
| usercategory = &lt;includeonly&gt;Wikipedians who read manga&lt;/includeonly&gt;
}}&lt;includeonly&gt;</text>
      <sha1>5v0kyowghfdih2td4xy75styqsipdx7</sha1>
    </revision>
  </page>
  <page>
    <title>Template:User/manga/User manga-</title>
    <ns>10</ns>
    <id>330129</id>
    <revision>
      <id>2815116</id>
      <timestamp>2026-06-10T20:57:39Z</timestamp>
      <contributor>
        <username>AUBSTRAWBS</username>
        <id>3060598</id>
      </contributor>
      <comment>Created page with &quot;{{userbox | border-c = #FFB3B3 | id       = [[:Category:Wikipedians interested in anime and manga|&gt;?&lt;]] | id-c     = #FFB3B3 | id-s     = 14 | info     = This person doesn't understand how to read manga (but may like to learn to) and requires counselling after every attempt of comprehension. | info-c   = #FFE0E8 | info-s   = 8 | info-lh  = 1.25em }}&lt;includeonly&gt;&quot;</comment>
      <origin>2815116</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="364" sha1="aqycwqbdp5eefzb8s0mmf6tt2hrzs2j" xml:space="preserve">{{userbox
| border-c = #FFB3B3
| id       = [[:Category:Wikipedians interested in anime and manga|&gt;?&lt;]]
| id-c     = #FFB3B3
| id-s     = 14
| info     = This person doesn't understand how to read manga (but may like to learn to) and requires counselling after every attempt of comprehension.
| info-c   = #FFE0E8
| info-s   = 8
| info-lh  = 1.25em
}}&lt;includeonly&gt;</text>
      <sha1>aqycwqbdp5eefzb8s0mmf6tt2hrzs2j</sha1>
    </revision>
  </page>
  <page>
    <title>Template:User/manga/User manga-o</title>
    <ns>10</ns>
    <id>330130</id>
    <revision>
      <id>2815117</id>
      <timestamp>2026-06-10T21:00:45Z</timestamp>
      <contributor>
        <username>AUBSTRAWBS</username>
        <id>3060598</id>
      </contributor>
      <comment>Created page with &quot;{{userbox | border-c = #FFB3B3 | id       = [[:Category:Wikipedians interested in anime and manga|&gt;_&lt;]] | id-c     = #FFB3B3 | id-s     = 14 | info     = This person '''dislikes''' manga (or doesn't admit to reading it). | info-c   = #FFE0E8 | info-s   = 8 | info-lh  = 1.25em }}&lt;noinclude&gt;&lt;br style=&quot;clear: both;&quot;&gt;&quot;</comment>
      <origin>2815117</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="315" sha1="jsnmcgeydaf27ouv17ra372kuxzi44n" xml:space="preserve">{{userbox
| border-c = #FFB3B3
| id       = [[:Category:Wikipedians interested in anime and manga|&gt;_&lt;]]
| id-c     = #FFB3B3
| id-s     = 14
| info     = This person '''dislikes''' manga (or doesn't admit to reading it).
| info-c   = #FFE0E8
| info-s   = 8
| info-lh  = 1.25em
}}&lt;noinclude&gt;&lt;br style=&quot;clear: both;&quot;&gt;</text>
      <sha1>jsnmcgeydaf27ouv17ra372kuxzi44n</sha1>
    </revision>
  </page>
  <page>
    <title>Template:User/manga/User manga-1</title>
    <ns>10</ns>
    <id>330131</id>
    <revision>
      <id>2815118</id>
      <timestamp>2026-06-10T21:03:03Z</timestamp>
      <contributor>
        <username>AUBSTRAWBS</username>
        <id>3060598</id>
      </contributor>
      <comment>Created page with &quot;{{userbox | border-c = #C0C8FF | id       = [[:Category:Wikipedians interested in anime and manga|._.]] | id-c     = #C0C8FF | id-s     = 14 | info     = This user sometimes reads manga. | info-c   = #F0F8FF | info-s   = 8 | info-lh  = 1.25em }}&lt;includeonly&gt;&lt;br style=&quot;clear: both;&quot;&gt;&quot;</comment>
      <origin>2815118</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="283" sha1="ixky3q099f5hrrw0ymeisxl7qbupq86" xml:space="preserve">{{userbox
| border-c = #C0C8FF
| id       = [[:Category:Wikipedians interested in anime and manga|._.]]
| id-c     = #C0C8FF
| id-s     = 14
| info     = This user sometimes reads manga.
| info-c   = #F0F8FF
| info-s   = 8
| info-lh  = 1.25em
}}&lt;includeonly&gt;&lt;br style=&quot;clear: both;&quot;&gt;</text>
      <sha1>ixky3q099f5hrrw0ymeisxl7qbupq86</sha1>
    </revision>
    <revision>
      <id>2815119</id>
      <parentid>2815118</parentid>
      <timestamp>2026-06-10T21:04:01Z</timestamp>
      <contributor>
        <username>AUBSTRAWBS</username>
        <id>3060598</id>
      </contributor>
      <origin>2815119</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="258" sha1="28qd33f6ipz550s1qkd9dyp2o5ruaxa" xml:space="preserve">{{userbox
| border-c = #C0C8FF
| id       = [[:Category:Wikipedians interested in anime and manga|._.]]
| id-c     = #C0C8FF
| id-s     = 14
| info     = This user sometimes reads manga.
| info-c   = #F0F8FF
| info-s   = 8
| info-lh  = 1.25em
}}&lt;includeonly&gt;</text>
      <sha1>28qd33f6ipz550s1qkd9dyp2o5ruaxa</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Userboxes/anime</title>
    <ns>4</ns>
    <id>330132</id>
    <revision>
      <id>2815121</id>
      <timestamp>2026-06-10T21:24:39Z</timestamp>
      <contributor>
        <username>AUBSTRAWBS</username>
        <id>3060598</id>
      </contributor>
      <comment>New anime userbox page</comment>
      <origin>2815121</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1624" sha1="abzcckwxen9hw9zxlnyaimdabja3tbp" xml:space="preserve">Hello and welcome to the Wikiversity '''[[anime]]''' '''[[userbox]]''' page.
This page currenly has a comrehensive list of all the anime/manga related userboxes on wikiversity. We are currently bringing over userboxes from wikipedia.

==Various==

===Anime===

{|
!Adding this to your page!!Creates this
|-
|&lt;nowiki&gt;{{User animanga userbox}}&lt;/nowiki&gt;||{{User animanga userbox}}
|-
|&lt;nowiki&gt;{{User anime user anime}}&lt;/nowiki&gt;||{{User anime user anime}}
|-
|&lt;nowiki&gt;{{User anime User anime-0}}&lt;/nowiki&gt;||{{User anime User anime-0}}
|-
|&lt;nowiki&gt;{{User anime User anime-1}}&lt;/nowiki&gt;||{{User anime User anime-1}}
|-
|&lt;nowiki&gt;{{User anime User anime-2}}&lt;/nowiki&gt;||{{User anime User anime-2}}
|-
|&lt;nowiki&gt;{{User anime User anime-3}}&lt;/nowiki&gt;||{{User anime User anime-3}}
|-
|&lt;nowiki&gt;{{User anime User anime-4}}&lt;/nowiki&gt;||{{User anime User anime-4}}
|-
|&lt;nowiki&gt;{{User/anime/User anime-N}}&lt;/nowiki&gt;||{{User/anime/User anime-N}}
|-
|&lt;nowiki&gt;{{Template:User/anime/Better actors}}&lt;/nowiki&gt;||{{Template:User/anime/Better actors}}
|}

=== Manga ===

{|
!Adding this to your page!!Creates this
|-
|&lt;nowiki&gt;{{User/manga/User manga}}&lt;/nowiki&gt;||{{User/manga/User manga}}
|-
|&lt;nowiki&gt;{{User/manga/User manga-}}&lt;/nowiki&gt;||{{User/manga/User manga-}}
|-
|&lt;nowiki&gt;{{User/manga/User manga-o}}&lt;/nowiki&gt;||{{User/manga/User manga-o}}
|-
|&lt;nowiki&gt;{{User/manga/User manga-1}}&lt;/nowiki&gt;||{{User/manga/User manga-1}}
|}

== Anime before it gets moved to better page ==

{|
!Adding this to your page!!Creates this
|-
|&lt;nowiki&gt;{{User gate anime}}&lt;/nowiki&gt;||{{User gate anime}}
|-
|&lt;nowiki&gt;{{User apothecary diaries}}&lt;/nowiki&gt;||{{User apothecary diaries}}
|}</text>
      <sha1>abzcckwxen9hw9zxlnyaimdabja3tbp</sha1>
    </revision>
  </page>
  <page>
    <title>Category:Motivation and emotion/Book/Self-blame</title>
    <ns>14</ns>
    <id>330133</id>
    <revision>
      <id>2815127</id>
      <timestamp>2026-06-10T22:11:06Z</timestamp>
      <contributor>
        <username>Jtneill</username>
        <id>10242</id>
      </contributor>
      <comment>Created page with &quot;[[Category:Motivation and emotion/Book/Self]]&quot;</comment>
      <origin>2815127</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45" sha1="ea3826x1qgjek0ssmbyjmbiuwkf1l3r" xml:space="preserve">[[Category:Motivation and emotion/Book/Self]]</text>
      <sha1>ea3826x1qgjek0ssmbyjmbiuwkf1l3r</sha1>
    </revision>
  </page>
  <page>
    <title>WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025</title>
    <ns>0</ns>
    <id>330134</id>
    <revision>
      <id>2815128</id>
      <timestamp>2026-06-10T22:12:13Z</timestamp>
      <contributor>
        <username>CorraleH</username>
        <id>2903442</id>
      </contributor>
      <comment>Proceedings of the WikiConf Colombia 2025</comment>
      <origin>2815128</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3126" sha1="qbna7wii8tjpowtvbblr05580g89y4h" xml:space="preserve">{{Article info
| first1 = Mónica
| last1 = Bonilla
| orcid1 = 0000-0002-4594-0093
| affiliation1 = Wikimedia Colombia
| first2 = Johana
| last2 = Botero
| orcid2 = 0009-0006-0484-6291
| affiliation2 = Wikimedia Colombia
| first3 = Bernardo
| last3 = Caycedo
| orcid3 = 0009-0008-1861-5034
| affiliation3 = Wikimedia Colombia
| first4 = Manuel
| last4 = Franco-Avellaneda
| orcid4 = 0000-0002-0895-8219
| affiliation4 = Wikimedia Colombia
| first5 = Nathaly
| last5 = Montoya
| orcid5 = 0009-0004-3539-5599
| affiliation5 = Wikimedia Colombia
| submitted=2025-11-14
| published = 2026-06-10
| correspondence1 = 
| journal = WikiJournal of Humanities
| w1 = 
| license = {{CC-BY-SA work}}
| abstract =
}}

[[File:Banner wikiconf wiki.png|thumb|alt=Wikiconferencia Colombia 2025|WikiConf Colombia 2025|link=https://meta.wikimedia.org/wiki/Wikiconferencia_Colombia_2025]]

==Foreword==
The Proceedings of the WikiConf Colombia 2025, held on November 14, 2025, in Bogotá, Colombia, bring together abstracts dedicated to the study of Wikipedia and other wiki projects as significant phenomena within the contemporary scientific, educational, and information space.

Organized by Wikimedia Colombia, with the support of the Wikimedia Foundation, the conference gathered participants from Colombia and abroad and marked an important step in fostering a scholarly community focused on interdisciplinary research of open knowledge and wiki environments. The contributions, published under the Creative Commons Attribution-ShareAlike 4.0 International license, reflect a wide range of research topics, including digital commons, free culture, historical memory, biodiversity, and Indigenous languages, while also emphasizing the role of Wikipedia in transforming prevailing digital paradigms and advancing the principles of free culture.

WikiConf Colombia 2025 constitutes the annual meeting of the Wikimedia community in Colombia. This year’s edition, held in Bogotá, aims to foster critical discussions on transforming prevailing digital paradigms and advancing the principles of free culture. The conference convenes a diverse range of participants, including local community members, Wikimedia Colombia (WMCO) staff, national scholarship recipients, and guests, to exchange experiences, address emerging challenges, and collaboratively envision more open and participatory futures for the Wikimedia movement in Colombia.

The conference seeks to reflect on digital commons as collaborative practices that challenge both the privatization of the digital sphere and uncritical forms of automation. It aims to open a space for imagining freer and more solidaristic futures in which open culture, historical memory, biodiversity, and Indigenous languages occupy a central place.

Digital commons are woven through diversity: they emerge in collaboration with Indigenous, Afro-descendant, rural, and urban communities, through the exchange between ancestral knowledge and open, emerging technologies. WikiConf Colombia 2025 serves as a moment of recognition, collective learning, and the creation of new alliances.

==Abstracts==

#</text>
      <sha1>qbna7wii8tjpowtvbblr05580g89y4h</sha1>
    </revision>
    <revision>
      <id>2815136</id>
      <parentid>2815128</parentid>
      <timestamp>2026-06-10T22:28:27Z</timestamp>
      <contributor>
        <username>CorraleH</username>
        <id>2903442</id>
      </contributor>
      <minor />
      <comment>Abstracts</comment>
      <origin>2815136</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8273" sha1="0pl0agfggksmxsdh3k6ebz694017qkr" xml:space="preserve">{{Article info
| first1 = Mónica
| last1 = Bonilla
| orcid1 = 0000-0002-4594-0093
| affiliation1 = Wikimedia Colombia
| first2 = Johana
| last2 = Botero
| orcid2 = 0009-0006-0484-6291
| affiliation2 = Wikimedia Colombia
| first3 = Bernardo
| last3 = Caycedo
| orcid3 = 0009-0008-1861-5034
| affiliation3 = Wikimedia Colombia
| first4 = Manuel
| last4 = Franco-Avellaneda
| orcid4 = 0000-0002-0895-8219
| affiliation4 = Wikimedia Colombia
| first5 = Nathaly
| last5 = Montoya
| orcid5 = 0009-0004-3539-5599
| affiliation5 = Wikimedia Colombia
| submitted = 2025-11-14
| published = 2026-06-10
| correspondence1 = 
| journal = WikiJournal of Humanities
| w1 = 
| license = {{CC-BY-SA work}}
| abstract =
}}

[[File:Banner wikiconf wiki.png|thumb|alt=Wikiconferencia Colombia 2025|WikiConf Colombia 2025|link=https://meta.wikimedia.org/wiki/Wikiconferencia_Colombia_2025]]

==Foreword==
The Proceedings of the WikiConf Colombia 2025, held on November 14, 2025, in Bogotá, Colombia, bring together abstracts dedicated to the study of Wikipedia and other wiki projects as significant phenomena within the contemporary scientific, educational, and information space.

Organized by Wikimedia Colombia, with the support of the Wikimedia Foundation, the conference gathered participants from Colombia and abroad and marked an important step in fostering a scholarly community focused on interdisciplinary research of open knowledge and wiki environments. The contributions, published under the Creative Commons Attribution-ShareAlike 4.0 International license, reflect a wide range of research topics, including digital commons, free culture, historical memory, biodiversity, and Indigenous languages, while also emphasizing the role of Wikipedia in transforming prevailing digital paradigms and advancing the principles of free culture.

WikiConf Colombia 2025 constitutes the annual meeting of the Wikimedia community in Colombia. This year’s edition, held in Bogotá, aims to foster critical discussions on transforming prevailing digital paradigms and advancing the principles of free culture. The conference convenes a diverse range of participants, including local community members, Wikimedia Colombia (WMCO) staff, national scholarship recipients, and guests, to exchange experiences, address emerging challenges, and collaboratively envision more open and participatory futures for the Wikimedia movement in Colombia.

The conference seeks to reflect on digital commons as collaborative practices that challenge both the privatization of the digital sphere and uncritical forms of automation. It aims to open a space for imagining freer and more solidaristic futures in which open culture, historical memory, biodiversity, and Indigenous languages occupy a central place.

Digital commons are woven through diversity: they emerge in collaboration with Indigenous, Afro-descendant, rural, and urban communities, through the exchange between ancestral knowledge and open, emerging technologies. WikiConf Colombia 2025 serves as a moment of recognition, collective learning, and the creation of new alliances.

==Abstracts==

# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Opening: Wikimedia Colombia and Universidad del Rosario|Opening: Wikimedia Colombia and Universidad del Rosario]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Roundtable: the commons in dispute: sovereignty and free knowledge from Wikimedia|Roundtable: the commons in dispute: sovereignty and free knowledge from Wikimedia]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Joo'uya Waashajaaiwaa Wanüiki - teaching Wayuunaiki with digital media|Joo'uya Waashajaaiwaa Wanüiki - teaching Wayuunaiki with digital media]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/How to build a Wikipedia article from zero and not die during the process? a situated experience|How to build a Wikipedia article from zero and not die during the process? a situated experience]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Free metaphor|Free metaphor]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Oral libraries in the midst of a knowledge system based on the written word|Oral libraries in the midst of a knowledge system based on the written word]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Roundtable: the path toward community environmental monitoring|Roundtable: the path toward community environmental monitoring]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Panel: AI and digital commons|Panel: AI and digital commons]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/We are not communists, we are commoners|We are not communists, we are commoners]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Decentralized Web3 governance for the Wikimedia community?|Decentralized Web3 governance for the Wikimedia community?]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/OpenStreetMap and Wikimedia: two worlds full of data that tell stories|OpenStreetMap and Wikimedia: two worlds full of data that tell stories]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Congressional and presidential elections are coming up: what can we do as the Wikimedia Colombia community?|Congressional and presidential elections are coming up: what can we do as the Wikimedia Colombia community?]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Wikimedia in the teaching of phylogenomics in Colombia|Wikimedia in the teaching of phylogenomics in Colombia]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Inclusion and sustainability at Wikimania Nairobi 2025: a narrative|Inclusion and sustainability at Wikimania Nairobi 2025: a narrative]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/#Fracking: tracing climate change disinformation on social media|#Fracking: tracing climate change disinformation on social media]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Roundtable: linguistic diversity and digital fabric|Roundtable: linguistic diversity and digital fabric]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Electoral API: how to transform public information into citizen knowledge? open data for democracy|Electoral API: how to transform public information into citizen knowledge? open data for democracy]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Stories, experiences, and images of the mothers of victims of false positives: process to build the memorial for women belonging to the MAFAPO collective|Stories, experiences, and images of the mothers of victims of false positives: process to build the memorial for women belonging to the MAFAPO collective]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/The river that we are|The river that we are]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Lightning talk: echoes of a mural|Lightning talk: echoes of a mural]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Restrictions and possibilities of Colombia's public [but not] open data|Restrictions and possibilities of Colombia's public [but not] open data]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Co-creation of a translation methodology using Wikibooks|Co-creation of a translation methodology using Wikibooks]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Superchange: the universe where climate heroes are born|Superchange: the universe where climate heroes are born]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Photographic co-creation workshop: self-representation and alterity in Abya Yala|Photographic co-creation workshop: self-representation and alterity in Abya Yala]]

&lt;!-- 
[[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Hackathon|Hackathon]]
[[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Poster session|Poster session]]
--&gt;</text>
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[[File:Banner wikiconf wiki.png|thumb|alt=Wikiconferencia Colombia 2025|WikiConf Colombia 2025|link=https://meta.wikimedia.org/wiki/Wikiconferencia_Colombia_2025]]

==Foreword==
The Proceedings of the WikiConf Colombia 2025, held on November 14, 2025, in Bogotá, Colombia, bring together abstracts dedicated to the study of Wikipedia and other wiki projects as significant phenomena within the contemporary scientific, educational, and information space.

Organized by Wikimedia Colombia, with the support of the Wikimedia Foundation, the conference gathered participants from Colombia and abroad and marked an important step in fostering a scholarly community focused on interdisciplinary research of open knowledge and wiki environments. The contributions, published under the Creative Commons Attribution-ShareAlike 4.0 International license, reflect a wide range of research topics, including digital commons, free culture, historical memory, biodiversity, and Indigenous languages, while also emphasizing the role of Wikipedia in transforming prevailing digital paradigms and advancing the principles of free culture.

WikiConf Colombia 2025 constitutes the annual meeting of the Wikimedia community in Colombia. This year’s edition, held in Bogotá, aims to foster critical discussions on transforming prevailing digital paradigms and advancing the principles of free culture. The conference convenes a diverse range of participants, including local community members, Wikimedia Colombia (WMCO) staff, national scholarship recipients, and guests, to exchange experiences, address emerging challenges, and collaboratively envision more open and participatory futures for the Wikimedia movement in Colombia.

The conference seeks to reflect on digital commons as collaborative practices that challenge both the privatization of the digital sphere and uncritical forms of automation. It aims to open a space for imagining freer and more solidaristic futures in which open culture, historical memory, biodiversity, and Indigenous languages occupy a central place.

Digital commons are woven through diversity: they emerge in collaboration with Indigenous, Afro-descendant, rural, and urban communities, through the exchange between ancestral knowledge and open, emerging technologies. WikiConf Colombia 2025 serves as a moment of recognition, collective learning, and the creation of new alliances.

==Abstracts==

# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Opening: Wikimedia Colombia and Universidad del Rosario|Opening: Wikimedia Colombia and Universidad del Rosario]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Roundtable: the commons in dispute: sovereignty and free knowledge from Wikimedia|Roundtable: the commons in dispute: sovereignty and free knowledge from Wikimedia]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Joo'uya Waashajaaiwaa Wanüiki - teaching Wayuunaiki with digital media|Joo'uya Waashajaaiwaa Wanüiki - teaching Wayuunaiki with digital media]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/How to build a Wikipedia article from zero and not die during the process? a situated experience|How to build a Wikipedia article from zero and not die during the process? a situated experience]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Free metaphor|Free metaphor]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Oral libraries in the midst of a knowledge system based on the written word|Oral libraries in the midst of a knowledge system based on the written word]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Roundtable: the path toward community environmental monitoring|Roundtable: the path toward community environmental monitoring]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Panel: AI and digital commons|Panel: AI and digital commons]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/We are not communists, we are commoners|We are not communists, we are commoners]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Decentralized Web3 governance for the Wikimedia community?|Decentralized Web3 governance for the Wikimedia community?]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/OpenStreetMap and Wikimedia: two worlds full of data that tell stories|OpenStreetMap and Wikimedia: two worlds full of data that tell stories]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Congressional and presidential elections are coming up: what can we do as the Wikimedia Colombia community?|Congressional and presidential elections are coming up: what can we do as the Wikimedia Colombia community?]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Wikimedia in the teaching of phylogenomics in Colombia|Wikimedia in the teaching of phylogenomics in Colombia]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Inclusion and sustainability at Wikimania Nairobi 2025: a narrative|Inclusion and sustainability at Wikimania Nairobi 2025: a narrative]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/#Fracking: tracing climate change disinformation on social media|#Fracking: tracing climate change disinformation on social media]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Roundtable: linguistic diversity and digital fabric|Roundtable: linguistic diversity and digital fabric]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Electoral API: how to transform public information into citizen knowledge? open data for democracy|Electoral API: how to transform public information into citizen knowledge? open data for democracy]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Stories, experiences, and images of the mothers of victims of false positives: process to build the memorial for women belonging to the MAFAPO collective|Stories, experiences, and images of the mothers of victims of false positives: process to build the memorial for women belonging to the MAFAPO collective]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/The river that we are|The river that we are]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Lightning talk: echoes of a mural|Lightning talk: echoes of a mural]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Restrictions and possibilities of Colombia's public (but not) open data|Restrictions and possibilities of Colombia's public (but not) open data]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Co-creation of a translation methodology using Wikibooks|Co-creation of a translation methodology using Wikibooks]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Superchange: the universe where climate heroes are born|Superchange: the universe where climate heroes are born]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Photographic co-creation workshop: self-representation and alterity in Abya Yala|Photographic co-creation workshop: self-representation and alterity in Abya Yala]]

&lt;!-- 
[[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Hackathon|Hackathon]]
[[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Poster session|Poster session]]
--&gt;</text>
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}}

[[File:Banner wikiconf wiki.png|thumb|alt=Wikiconferencia Colombia 2025|WikiConf Colombia 2025|link=https://meta.wikimedia.org/wiki/Wikiconferencia_Colombia_2025]]

==Foreword==
The Proceedings of the WikiConf Colombia 2025, held on November 14, 2025, in Bogotá, Colombia, bring together abstracts dedicated to the study of Wikipedia and other wiki projects as significant phenomena within the contemporary scientific, educational, and information space.

Organized by Wikimedia Colombia, with the support of the Wikimedia Foundation, the conference gathered participants from Colombia and abroad and marked an important step in fostering a scholarly community focused on interdisciplinary research of open knowledge and wiki environments. The contributions, published under the Creative Commons Attribution-ShareAlike 4.0 International license, reflect a wide range of research topics, including digital commons, free culture, historical memory, biodiversity, and Indigenous languages, while also emphasizing the role of Wikipedia in transforming prevailing digital paradigms and advancing the principles of free culture.

WikiConf Colombia 2025 constitutes the annual meeting of the Wikimedia community in Colombia. This year’s edition, held in Bogotá, aims to foster critical discussions on transforming prevailing digital paradigms and advancing the principles of free culture. The conference convenes a diverse range of participants, including local community members, Wikimedia Colombia (WMCO) staff, national scholarship recipients, and guests, to exchange experiences, address emerging challenges, and collaboratively envision more open and participatory futures for the Wikimedia movement in Colombia.

The conference seeks to reflect on digital commons as collaborative practices that challenge both the privatization of the digital sphere and uncritical forms of automation. It aims to open a space for imagining freer and more solidaristic futures in which open culture, historical memory, biodiversity, and Indigenous languages occupy a central place.

Digital commons are woven through diversity: they emerge in collaboration with Indigenous, Afro-descendant, rural, and urban communities, through the exchange between ancestral knowledge and open, emerging technologies. WikiConf Colombia 2025 serves as a moment of recognition, collective learning, and the creation of new alliances.

==Abstracts==

# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Opening: Wikimedia Colombia and Universidad del Rosario|Opening: Wikimedia Colombia and Universidad del Rosario]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Roundtable: the commons in dispute: sovereignty and free knowledge from Wikimedia|Roundtable: the commons in dispute: sovereignty and free knowledge from Wikimedia]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Joo'uya Waashajaaiwaa Wanüiki - teaching Wayuunaiki with digital media|Joo'uya Waashajaaiwaa Wanüiki - teaching Wayuunaiki with digital media]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/How to build a Wikipedia article from zero and not die during the process? a situated experience|How to build a Wikipedia article from zero and not die during the process? a situated experience]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Free metaphor|Free metaphor]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Oral libraries in the midst of a knowledge system based on the written word|Oral libraries in the midst of a knowledge system based on the written word]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Roundtable: the path toward community environmental monitoring|Roundtable: the path toward community environmental monitoring]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Panel: AI and digital commons|Panel: AI and digital commons]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/We are not communists, we are commoners|We are not communists, we are commoners]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Decentralized Web3 governance for the Wikimedia community?|Decentralized Web3 governance for the Wikimedia community?]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/OpenStreetMap and Wikimedia: two worlds full of data that tell stories|OpenStreetMap and Wikimedia: two worlds full of data that tell stories]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Congressional and presidential elections are coming up: what can we do as the Wikimedia Colombia community?|Congressional and presidential elections are coming up: what can we do as the Wikimedia Colombia community?]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Wikimedia in the teaching of phylogenomics in Colombia|Wikimedia in the teaching of phylogenomics in Colombia]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Inclusion and sustainability at Wikimania Nairobi 2025: a narrative|Inclusion and sustainability at Wikimania Nairobi 2025: a narrative]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/#Fracking: tracing climate change disinformation on social media|#Fracking: tracing climate change disinformation on social media]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Roundtable: linguistic diversity and digital fabric|Roundtable: linguistic diversity and digital fabric]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Electoral API: how to transform public information into citizen knowledge? open data for democracy|Electoral API: how to transform public information into citizen knowledge? open data for democracy]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Stories, experiences, and images of the mothers of victims of false positives: process to build the memorial for women belonging to the MAFAPO collective|Stories, experiences, and images of the mothers of victims of false positives: process to build the memorial for women belonging to the MAFAPO collective]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/The river that we are|The river that we are]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Lightning talk: echoes of a mural|Lightning talk: echoes of a mural]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Restrictions and possibilities of Colombia's public (but not) open data|Restrictions and possibilities of Colombia's public (but not) open data]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Co-creation of a translation methodology using Wikibooks|Co-creation of a translation methodology using Wikibooks]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Superchange: the universe where climate heroes are born|Superchange: the universe where climate heroes are born]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Photographic co-creation workshop: self-representation and alterity in Abya Yala|Photographic co-creation workshop: self-representation and alterity in Abya Yala]]

&lt;!-- 
[[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Hackathon|Hackathon]]
[[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Poster session|Poster session]]
--&gt;</text>
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}}

[[File:Banner wikiconf wiki.png|thumb|alt=Wikiconferencia Colombia 2025|Wikiconferencia Colombia 2025|link=https://meta.wikimedia.org/wiki/Wikiconferencia_Colombia_2025|320px]]

==Foreword==
The Proceedings of the [[meta:Wikiconferencia_Colombia_2025|WikiConf Colombia 2025]], held on November 14, 2025, in Bogotá, Colombia, bring together abstracts dedicated to the study of Wikipedia and other wiki projects as significant phenomena within the contemporary scientific, educational, and information space.

Organized by Wikimedia Colombia, with the support of the Wikimedia Foundation, the conference gathered participants from Colombia and abroad and marked an important step in fostering a scholarly community focused on interdisciplinary research of open knowledge and wiki environments. The contributions, published under the Creative Commons Attribution-ShareAlike 4.0 International license, reflect a wide range of research topics, including digital commons, free culture, historical memory, biodiversity, and Indigenous languages, while also emphasizing the role of Wikipedia in transforming prevailing digital paradigms and advancing the principles of free culture.

WikiConf Colombia 2025 constitutes the annual meeting of the Wikimedia community in Colombia. This year’s edition, held in Bogotá, aims to foster critical discussions on transforming prevailing digital paradigms and advancing the principles of free culture. The conference convenes a diverse range of participants, including local community members, Wikimedia Colombia (WMCO) staff, national scholarship recipients, and guests, to exchange experiences, address emerging challenges, and collaboratively envision more open and participatory futures for the Wikimedia movement in Colombia.

The conference seeks to reflect on digital commons as collaborative practices that challenge both the privatization of the digital sphere and uncritical forms of automation. It aims to open a space for imagining freer and more solidaristic futures in which open culture, historical memory, biodiversity, and Indigenous languages occupy a central place.

Digital commons are woven through diversity: they emerge in collaboration with Indigenous, Afro-descendant, rural, and urban communities, through the exchange between ancestral knowledge and open, emerging technologies. WikiConf Colombia 2025 serves as a moment of recognition, collective learning, and the creation of new alliances.

==Abstracts==

# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Opening: Wikimedia Colombia and Universidad del Rosario|Opening: Wikimedia Colombia and Universidad del Rosario]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Roundtable: the commons in dispute: sovereignty and free knowledge from Wikimedia|Roundtable: the commons in dispute: sovereignty and free knowledge from Wikimedia]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Joo'uya Waashajaaiwaa Wanüiki - teaching Wayuunaiki with digital media|Joo'uya Waashajaaiwaa Wanüiki - teaching Wayuunaiki with digital media]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/How to build a Wikipedia article from zero and not die during the process? a situated experience|How to build a Wikipedia article from zero and not die during the process? a situated experience]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Free metaphor|Free metaphor]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Oral libraries in the midst of a knowledge system based on the written word|Oral libraries in the midst of a knowledge system based on the written word]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Roundtable: the path toward community environmental monitoring|Roundtable: the path toward community environmental monitoring]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Panel: AI and digital commons|Panel: AI and digital commons]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/We are not communists, we are commoners|We are not communists, we are commoners]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Decentralized Web3 governance for the Wikimedia community?|Decentralized Web3 governance for the Wikimedia community?]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/OpenStreetMap and Wikimedia: two worlds full of data that tell stories|OpenStreetMap and Wikimedia: two worlds full of data that tell stories]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Congressional and presidential elections are coming up: what can we do as the Wikimedia Colombia community?|Congressional and presidential elections are coming up: what can we do as the Wikimedia Colombia community?]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Wikimedia in the teaching of phylogenomics in Colombia|Wikimedia in the teaching of phylogenomics in Colombia]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Inclusion and sustainability at Wikimania Nairobi 2025: a narrative|Inclusion and sustainability at Wikimania Nairobi 2025: a narrative]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/#Fracking: tracing climate change disinformation on social media|#Fracking: tracing climate change disinformation on social media]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Roundtable: linguistic diversity and digital fabric|Roundtable: linguistic diversity and digital fabric]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Electoral API: how to transform public information into citizen knowledge? open data for democracy|Electoral API: how to transform public information into citizen knowledge? open data for democracy]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Stories, experiences, and images of the mothers of victims of false positives: process to build the memorial for women belonging to the MAFAPO collective|Stories, experiences, and images of the mothers of victims of false positives: process to build the memorial for women belonging to the MAFAPO collective]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/The river that we are|The river that we are]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Lightning talk: echoes of a mural|Lightning talk: echoes of a mural]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Restrictions and possibilities of Colombia's public (but not) open data|Restrictions and possibilities of Colombia's public (but not) open data]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Co-creation of a translation methodology using Wikibooks|Co-creation of a translation methodology using Wikibooks]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Superchange: the universe where climate heroes are born|Superchange: the universe where climate heroes are born]]
# [[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Photographic co-creation workshop: self-representation and alterity in Abya Yala|Photographic co-creation workshop: self-representation and alterity in Abya Yala]]

&lt;!-- 
[[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Hackathon|Hackathon]]
[[WikiJournal of Humanities/Proceedings of the WikiConf Colombia 2025/Poster session|Poster session]]
--&gt;</text>
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