Wikiversity:Colloquium

<mediawiki xmlns="http://www.mediawiki.org/xml/export-0.11/" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://www.mediawiki.org/xml/export-0.11/ http://www.mediawiki.org/xml/export-0.11.xsd" version="0.11" xml:lang="en">
  <siteinfo>
    <sitename>Wikiversity</sitename>
    <dbname>enwikiversity</dbname>
    <base>https://en.wikiversity.org/wiki/Wikiversity:Main_Page</base>
    <generator>MediaWiki 1.47.0-wmf.1</generator>
    <case>first-letter</case>
    <namespaces>
      <namespace key="-2" case="first-letter">Media</namespace>
      <namespace key="-1" case="first-letter">Special</namespace>
      <namespace key="0" case="first-letter" />
      <namespace key="1" case="first-letter">Talk</namespace>
      <namespace key="2" case="first-letter">User</namespace>
      <namespace key="3" case="first-letter">User talk</namespace>
      <namespace key="4" case="first-letter">Wikiversity</namespace>
      <namespace key="5" case="first-letter">Wikiversity talk</namespace>
      <namespace key="6" case="first-letter">File</namespace>
      <namespace key="7" case="first-letter">File talk</namespace>
      <namespace key="8" case="first-letter">MediaWiki</namespace>
      <namespace key="9" case="first-letter">MediaWiki talk</namespace>
      <namespace key="10" case="first-letter">Template</namespace>
      <namespace key="11" case="first-letter">Template talk</namespace>
      <namespace key="12" case="first-letter">Help</namespace>
      <namespace key="13" case="first-letter">Help talk</namespace>
      <namespace key="14" case="first-letter">Category</namespace>
      <namespace key="15" case="first-letter">Category talk</namespace>
      <namespace key="100" case="first-letter">School</namespace>
      <namespace key="101" case="first-letter">School talk</namespace>
      <namespace key="102" case="first-letter">Portal</namespace>
      <namespace key="103" case="first-letter">Portal talk</namespace>
      <namespace key="104" case="first-letter">Topic</namespace>
      <namespace key="105" case="first-letter">Topic talk</namespace>
      <namespace key="106" case="first-letter">Collection</namespace>
      <namespace key="107" case="first-letter">Collection talk</namespace>
      <namespace key="118" case="first-letter">Draft</namespace>
      <namespace key="119" case="first-letter">Draft talk</namespace>
      <namespace key="710" case="first-letter">TimedText</namespace>
      <namespace key="711" case="first-letter">TimedText talk</namespace>
      <namespace key="828" case="first-letter">Module</namespace>
      <namespace key="829" case="first-letter">Module talk</namespace>
      <namespace key="1728" case="first-letter">Event</namespace>
      <namespace key="1729" case="first-letter">Event talk</namespace>
    </namespaces>
  </siteinfo>
  <page>
    <title>Wikiversity:Colloquium</title>
    <ns>4</ns>
    <id>28</id>
    <revision>
      <id>2807809</id>
      <parentid>2807796</parentid>
      <timestamp>2026-05-06T13:32:25Z</timestamp>
      <contributor>
        <username>Codename Noreste</username>
        <id>2969951</id>
      </contributor>
      <minor />
      <comment>/* Interface administrator for Codename Noreste */ Format. (using [[wikt:MediaWiki:Gadget-AjaxEdit.js|AjaxEdit]])</comment>
      <origin>2807809</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="76052" sha1="kpsfukxajaj8gpft4lt37bcjipwful2" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;
== Requested update to [[Wikiversity:Interface administrators]] ==

Currently, [[Wikiversity:Interface administrators]] is a policy that includes a caveat that interface admins are not required long-term and that user right can only be added for a period of up to two weeks. I am proposing that we remove this qualification and allow for indefinite interface admin status. I think this is useful because there are reasons for tweaking the site CSS or JavaScript (e.g. to comply with dark mode), add gadgets (e.g. importing Cat-a-Lot, which I would like to do), or otherwise modifying the site that could plausibly come up on an irregular basis and requiring the overhead of a bureaucrat to add the user rights is inefficient. In particular, I am also going to request this right if the community accepts indefinite interface admins. Thoughts? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:23, 17 August 2025 (UTC)

:And who will then monitor them to make sure they don't damage the project in any way, or abuse the rights acquired in this way? For large projects, this might not be a problem, but for smaller projects like the English Wikiversity, I'm not sure if there are enough users who would say, something is happening here that shouldn't be happening. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:28, 20 August 2025 (UTC)
::Anyone would be who. This argument applies to any person with any advanced rights here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 10:46, 20 August 2025 (UTC)
:I think it is reasonable to allow for longer periods of access than 2 weeks to interface admin and support adjusting the policy to allow for this flexibility. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:57, 2 December 2025 (UTC)
::+1 —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:38, 25 January 2026 (UTC)
:@[[User:Koavf|Koavf]] I agree that the two-week requirement could be revised, but wouldn’t people just request access for a specific purpose anyway? Instead of granting indefinite access, they should request the specific time frame they need the rights for—until the planned fixes are completed—and then request an extension if more time is required. We could remove the two-week criterion while still keeping the access explicitly temporary. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:48, 25 January 2026 (UTC)
::I just don't see why this wiki needs to be different than all of the others. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:18, 25 January 2026 (UTC)
:::There isn’t really much of a need for a permanent one at this point in time [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:53, 25 January 2026 (UTC)
:I quite agree with this proposal, so long as they perform the suggested changes as mentioned here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 04:06, 26 January 2026 (UTC)
:: Just to clarify, I support '''indefinite interface admin status'''. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:34, 13 April 2026 (UTC)
:I think there is decent consensus for lengthening this, but not necessarily for indefinite permissions, so does anyone object to me revising it to the standard being 120 days instead of two weeks? I'll check back on this thread in three weeks and if there's no objection, I'll make the change. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:47, 13 April 2026 (UTC)
::Sure [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:27, 13 April 2026 (UTC)
::Thanks for proposing this, Justin. I agree with the proposal to lengthen the interface admin period from 2 weeks but not indefinitely. Can I check the source(s) for the standard being 120 days (I'm guessing policies on other projects or maybe global policy?)? In any case, I think it is reasonable for us to adopt a similar period. However, note on the current policy discussion page notes from @[[User:Dave Braunschweig|Dave Braunschweig]] arguing for shorter periods to lower risk, that's why it is 2 weeks. But if there are projects that need longer access, that should also be accommodated. Maybe we could adjust the policy to specify that ''interface admin rights can be given for 14 to 120 days depending on how long is required and what is supported by the community''. Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:29, 24 April 2026 (UTC)
:::There was there was no source for 120: it was just more than 14 and less than infinity. The &quot;14 to 120&quot; also seems reasonable. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 14:33, 24 April 2026 (UTC)
::: On some small/medium-sized wikis, such as English Wikibooks and English Wikiquote for example, indefinite interface administrator access for administrators is allowed, but they tend not to make changes to the CSS and JS page changes unless it's truly necessary. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:34, 24 April 2026 (UTC)
:::It's a good idea to make the length of this right on request or allow to be prolonged. However, IA should test large changes somewhere else, for example on the en.wv mirror, and only after testing it on the mirror, adapt it to the live version. That means I can't imagine a time-consuming operation right now. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:04, 24 April 2026 (UTC)
::::Sorry, what mirror is this? Are you talking about beta.wv? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:32, 24 April 2026 (UTC)
:::::Not beta.wv. Basically somewhere else then on a live wiki. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:59, 24 April 2026 (UTC)
:::::: Wouldn't testing on a user's own common.css page work anyway? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:36, 24 April 2026 (UTC)
:Change made here: https://en.wikiversity.org/w/index.php?title=Wikiversity%3AInterface_administrators&amp;diff=2807543&amp;oldid=2806289 —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:35, 4 May 2026 (UTC)

== [[Template:AI-generated]] ==

After going through the plethora of ChatGPT-generated pages made by [[User:Lbeaumont|Lbeaumont]] (with many more pages to go), I'd like community input on this proposal to [[Wikiversity:Artificial intelligence]] that I think would be benefical for the community:
*Resources generated by AI '''must''' be indicated as so through the project box, [[Template:AI-generated]], on either the page or the main resource (if the page is a part of a project).
I do not believe including a small note/reference that a page is AI-generated is sufficient, and I take my thinking from [[WV:Original research|Wikiversity's OR policy]] for OR work: ''Within Wikiversity, all original research should be clearly identified as such''. I believe resources created from AI should also be clearly indicated as such, especially since we are working on whether or not AI-generated resources should be allowed on the website (discussion is [[Wikiversity talk:Artificial intelligence|here]], for reference). This makes it easier for organizational purposes, and in the event ''if'' we ban AI-generated work. 

I've left a message on Lee's talk page over a week ago and did not get a response or acknowledgement, so I'd like for the community's input for this inclusion to the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:53, 26 January 2026 (UTC)

:I believe that existing Wikiversity policies are sufficient. Authors are responsible for the accuracy and usefulness of the content that is published. This policy covers AI-generated content that is: 1) carefully reviewed by the author publishing it, and 2) the source is noted.   [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 19:38, 27 January 2026 (UTC)
::A small reference for pages that are substantially filled with Chat-GPT entries, like [[Real Good Religion]], [[Attributing Blame]], [[Fostering Curiosity]], are not sufficient IMO and a project box would be the best indicator that a page is AI-generated (especially when there is a mixture of human created content AND AI-generated content, as present in a lot of your pages). This is useful, especially considering the notable issues with AI (including hallucinations and fabrication of details), so viewers and support staff are aware. These small notes left on the pages are not as easily viewable as a project box or banner would be. I really don't see the issue with a clear-label guideline. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:34, 27 January 2026 (UTC)
::{{ping|Lbeaumont}} I noticed your reversions [https://en.wikiversity.org/w/index.php?title=Exploring_Existential_Concerns&amp;diff=prev&amp;oldid=2788278 here] &amp; [https://en.wikiversity.org/w/index.php?title=Subjective_Awareness&amp;diff=prev&amp;oldid=2788257 here]. I'd prefer to have a clean conversation regarding this proposition. Please voice your concerns here. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:53, 28 January 2026 (UTC)
:::Regarding Subjective Awareness, I distinctly recall the effort I went to to write that the old-fashioned way. It is true that ChatGPT assisted me in augmenting the list of words suggested as candidate subjective states. This is a small section of the course, is clearly marked, and makes no factual claim. Marking the entire course as AI-generated is misleading. I would have made these comments when I reverted your edit; however, the revert button does not provide that opportunity.
:::Regarding the Exploring Existential Concerns course, please note this was adapted from my EmotionalCompetency.com website, which predates the availability of LLMs. The course does include two links, clearly labeled as ChatGPT-generated. Again, marking the entire course as AI-generated is misleading.
:::On a broader issue, I don't consider your opinions to have established a carefully debated and adopted Wikiversity policy. You went ahead and modified many of my courses over my clearly stated objections. Please let this issue play out more completely before editing my courses further. Thanks.   [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 15:11, 29 January 2026 (UTC)
::::Understood, and I respect your position. I apologize if my edits were seen as overarching. We could change the project box to &quot;a portion of this resource was generated by AI&quot;, or something along those lines. Feel free to revert my changes where you see fit, and I encourage more users to provide their input. EDIT: I've made changes to the template to indicate that a portion of the content has been generated from an LLM. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:50, 29 January 2026 (UTC)
:::::Thanks for this reply. The new banner is unduly large and alarming. There is no need for alarm here. The use of AI is not harmful per se. Like any technology, it can be used to help or to harm. I take care to craft prompts carefully, point the LMM to reliable source materials, and to carefully read and verify the generated text before I publish it. This is all in keeping with long-established Wikiversity policy. We don't want to use a  [[w:One-drop_rule|one-drop rule]] here or cause a [[w:Satanic_panic|satanic panic]]. We can learn our lessons from history here. I don't see any pedagogical reason for establishing a classification of &quot;AI generated&quot;, but if there is a consensus that it is needed, perhaps it can be handled as just another category that learning resources can be assigned to. I would rather focus on identifying any errors in factual claims than on casting pejorative bias toward AI-generated content. An essay on the best practices for using LMM on Wikiveristy would be welcome.   [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 15:58, 30 January 2026 (UTC)
::::::The new banner mimics the banner that is available on the English Wikibooks (see [[b:Template:AI-generated]] &amp; [[b:Template:Uses AI]]), so my revisions aren't unique in this aspect. At this point, I'd welcome other peoples' inputs. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:40, 30 January 2026 (UTC)

== How do I start making pages? ==
Is there a notability guideline for Wikiversity? What is the sourcing policy for information? What is the Manual of Style? What kind of educational content qualifies for Wikiversity? All the introduction pages are a bit unclear. 
[[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 02:25, 28 January 2026 (UTC)

:{{ping|VidanaliK}} Welcome to Wikiversity! I've left you a welcome message on your talk page. That should help you out. Make sure to especially look at [[Wikiversity:Introduction]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 03:11, 28 January 2026 (UTC)
::It says that I can't post more pages because I have apparently exceeded the new page limit. How long does it take before that new page limit expires? [[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 16:57, 28 January 2026 (UTC)
:::This is a restriction for new users so that Wikiversity is not hit with massive spam. As for when this limit will expire, it should be a few days or after a certain number of edits. It's easy to overcome, though I do not have the exact numbers atm. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:08, 29 January 2026 (UTC)
::::OK, I think I got past the limit. [[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 17:21, 29 January 2026 (UTC)
==Why does it feel like Wikiversity is no longer really active anymore?==
I've been looking at recent changes, and both today and yesterday there haven't been many changes that I haven't made; it feels like walking through a ghost town, is this just me or is Wikiversity not really active anymore? [[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 03:54, 30 January 2026 (UTC)

:There is fewer people editing these days compared to the past. Many newcomers tend to edit in Wikipedia instead. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 06:39, 30 January 2026 (UTC)
:It’s a little slow, but I’m happy to know that Wikiversity is a place that I think should provide value even if the activity of editors fluctuates. If it’s any consolation your edits may be encouraging for some anonymous newcomer to start edits on their own! I think it’s hard to build community when there is such a wide variety of interests and a smaller starting userbase. Also sometimes the getting into a particular topic that already exists can be intimidating because some relics (large portals, school, categories, etc.) have intricate, unique and generally messy levels of organization. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 22:16, 9 March 2026 (UTC)
:I'd say it comes down to working hard for Wikiversity, basically if somebody or a group of people will start presenting good ideas and they turn out to be provably stable.
:I even asked Google's &quot;AI Mode&quot;, what is Wikiversity famous for? Unfortunately it could not answer that.
:Simply, we have not made Wikiversity famous by presenting really provable stable ideas yet. My hope is that this time might come. Perhaps even this year 2026!
:Hope dies last. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:12, 27 April 2026 (UTC)

== Inactivity policy for Curators ==

I was wondering if there is a specific inactivity polity for curators (semi-admins) as I am pretty sure the global policy does not apply to them as they are not ''fully'' sysops. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:20, 15 February 2026 (UTC)

:Unfortunately, I don't see an inactivity policy, but if we were to create such a new policy for curators, it should be the same for custodians (administrators). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:45, 15 February 2026 (UTC)
::@[[User:Codename Noreste|Codename Noreste]] There is currently none, that I could find, for custodians either. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:47, 17 February 2026 (UTC)
:::I think we should propose a local inactivity policy for custodians (and by extension, curators), which should be at least one year without any edits ''and'' logged actions. However, I don't know which page should it be when the inactivity removal procedure starts. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:53, 17 February 2026 (UTC)
::::@[[User:Codename Noreste|Codename Noreste]] In theory, there should be a section added at [[WV:Candidates for custodianship]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:55, 17 February 2026 (UTC)
::::: To be consistent with the [[meta:Admin activity review|global period of 2 years inactivity]] for en.wv  [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship#How are bureaucrats removed?|Bureaucrats]] we could add something like this to [[Wikiversity:Curators]]:
::::::The maximum time period of inactivity &lt;u&gt;without community review&lt;/u&gt; for curators is two years (consistent with  the [[:meta:Category:Global policies|global policy]] described at [[meta:Admin activity review|Admin activity review]] which applies for [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship|Bureaucrats]]). After that time a custodian will remove the rights.
::::: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:51, 27 March 2026 (UTC)
:::::Yup, I agree with Jtneill, there is a policy proposal for Wikiversity:Curators, where it should be logically deployed. The question is if we are ready to aprove the policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:43, 17 April 2026 (UTC)
:::::: I agree, but we should notify the colloquium about inactive curators, just like a steward would do for inactive custodians and bureaucrats per [[:m:Admin activity review|AAR]]. What is the minimum timeframe an inactive curator should receive so they can respond they would keep their rights? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:49, 17 April 2026 (UTC)
:I incorporated these suggestions into the proposed curators policy. Please review/comment/improve. Summary: 2 years, notify curator's user page, then remove rights after 1 month: [[Wikiversity:Curators#Inactivity]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:59, 24 April 2026 (UTC)
:: @[[User:Jtneill|Jtneill]] I created [[Template:Inactive curator]] for this. Feel free to make any changes or improvements. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:29, 24 April 2026 (UTC)

== [[Wikiversity:Artificial intelligence]] to become an official policy ==
{{Archive top|After running for a week, there is consensus, alongside comments, for [[Wikiversity:Artificial intelligence]] to be implemented as an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:27, 17 April 2026 (UTC)}}
With the introduction of AI-material, and some material just plain disruptive, its imperative that Wikiversity catches up with its sister projects and implements an official AI policy that we can work with. The recent issue of [[User:Lbeaumont|Lbeaumont]]'s 50+ articles that contain significantly large AI-generated material has made me came to the Colloquium. This user has also been removing the [[Template:AI-generated]] template from their pages, calling it &quot;misleading&quot;, &quot;alarmist&quot;, and &quot;pejorative&quot; - which is all just simply nonsensical rationales. Not to even mention this user's contributions to the English Wikipedia have been [https://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Inner_Development_Goals contested] and [https://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Multipolar_trap removed] a couple of times (for being low-quality and clearly LLM-generated), highlighting the need for an actual policy to be implemented here on Wikiversity. I would like to ping {{ping|Juandev}} and {{ping|Jtneill}} for their thoughts as well, since I'd like this to be implemented as soon as possible. 

Wikiversity has a significant issue with implementing anti-disruptive measures, hence why we have received numerous complaints as a community about our quality. I originally was reverting the removal of the templates, but realized that this is still a proposed policy, which it shouldn't be anymore. It should be a recognized Wikiversity policy. 14:54, 10 March 2026 (UTC) —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:54, 10 March 2026 (UTC)

:@[[User:Atcovi|Atcovi]] '''I agree''' that the draft, should become official policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:00, 10 March 2026 (UTC)
:I provided a detailed response at: [[Wikiversity talk:Artificial intelligence#Evolving a Wikiversity policy on AI]]
:I will appreaciate it if you consder that carefully. [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 22:49, 10 March 2026 (UTC)
:Agree it should become official Wikiversity policy on the condition &lt;u&gt;that point point 5 is about [significant/substantial] LLM-generated text specifically&lt;/u&gt;. Not a good idea to overuse it, it should be added when there is substantial AI-generated text on the page, not for other cases. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:37, 11 March 2026 (UTC)
:What policy is being debated? Is it the text on this page, which is pointed to by the general banner, or the text at:   [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence,]]   which is pointed to by the specific banner? Let's begin with coherence on the text being debated. Thanks!  [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 11:49, 17 March 2026 (UTC)
::@[[User:Lbeaumont|Lbeaumont]] This is a call for approval of the new Wikiversity policy. You expressed your opinion [[Wikiversity talk:Artificial intelligence#Evolving a Wikiversity policy on AI|on the talk page of the proposal]], I replied to you and await your response.When creating policies, it is necessary to propose specific solutions. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:12, 17 March 2026 (UTC)
:::Toward a Justified and Parsimonious AI Policy 
:::As we collaborate to develop a consensus policy on the use of Large Language Models, it is wise to begin by considering the needs of the various stakeholders to the policy.
:::The stakeholders are:
:::1)     The users, 
:::2)     The source providers, and
:::3)     The editors
:::There may also be others with a minor stake in this policy, including the population at large.
:::The many needs of the users are currently addressed by long-standing [[Wikiversity:Policies|Wikiversity policies]], so we can focus on what, if any, additional needs arise as LLMs are deployed. 
:::As always, users need assurance that propositional statements are accurate. This is covered by the existing policy on [[Wikiversity:Verifiability|verifiably]]. In addition, it is expected by both the users and those that provide materials used as sources for the text are [[Wikiversity:Cite sources|accurately attributed]]. This is also covered by [[Wikiversity:Cite sources|existing policies]].
:::To respect the time and effort of editors, a parsimonious policy will unburden editors from costly requirements that exceed benefits to the users.
:::Finally, it is important to recognize that because attention is our most valuable seizing attention unnecessarily is a form of theft. 
:::The following proposed policy statement results from these considerations:
:::Recommended Policy statement:
:::·       Editors [[Wikiversity:Verifiability|verify the accuracy]] of propositional statements, regardless of the source.
:::·       Editors [[Wikiversity:Cite sources|attribute the source]] of propositional statements. In the case of LLM, cite the LLM model and the prompt used.
:::·       Use of various available templates to mark the use of LLM are optional. Templates that are flexible in noting the type and extend of LLM usage are preferred. Templates that avoid unduly distracting or alarming the user are preferred. [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 19:56, 19 March 2026 (UTC)
::::Do we discuss here or there? I have replied you there as your proposal is about that policy so it is tradition to discuss it at the affected talk page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:59, 19 March 2026 (UTC)
: {{support}} Thanks for the proposed policy development and discussion; also note proposed policy talk page discussion: [[Wikiversity talk:Artificial intelligence]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:05, 24 March 2026 (UTC)
::I think the Wikiversity AI policy shall be official. – [[User:RestoreAccess111|RestoreAccess111]] &lt;sup style=&quot;font-family:Arimo, Arial;&quot;&gt;[[User talk:RestoreAccess111|Talk!]]&lt;/sup&gt; &lt;sup style=&quot;font-family:Times New Roman, Tinos;&quot;&gt;[[Special:Contributions/RestoreAccess111|Watch!]]&lt;/sup&gt; 06:11, 13 April 2026 (UTC)
{{archive bottom}}

== New titles for user right nominations ==
&lt;div class=&quot;cd-moveMark&quot;&gt;''Moved from [[Wikiversity talk:Candidates for Custodianship#New titles for user right nominations]].  [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:20, 17 April 2026 (UTC)''&lt;/div&gt;

I would like to propose the following retitles should a user be nominated for any of the following user rights:
* Curator: Candidates for Curatorship
* Bureaucrat: Candidates for Bureaucratship
The reason is that many curator (and probably bureaucrat) requests have run solely under {{tq|Candidates for Custodianship}}, but that title might sound misleading (especially in regards to the permission a user is requesting). CheckUser and Oversight (suppressor) are not included above since no user was nominated for these sensitive permissions, probably. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:30, 19 March 2026 (UTC)

:And it's not that when someone at the beginning misplaced the request, no one thought to move it and the others copied it. Even today, it would be possible to simply take it all and move it. Otherwise, for me, the more fundamental problem is that there is [[Wikiversity:Curators|no approved policy for curators]] than where the requests are based. Curators then operate in a certain vacuum and if one of them &quot;breaks out of the chain&quot;, the average user doesn't have many transparent tools to deal with it, because there is no policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:02, 19 March 2026 (UTC)
::I am not talking about the curator page (policy proposal). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:08, 21 March 2026 (UTC)
: @[[User:Juandev|Juandev]] I'll see if I can do an overhaul of [[Wikiversity:Candidates for Custodianship]], just like I recently did with the Requests for adminship page on English Wikiquote. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:17, 18 April 2026 (UTC)
:Yes, great idea - ideally there will be separate &quot;Candidates for ...&quot; pages for each user right group. The most important for now is to separate curator and custodian pages as CN suggests. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 1 May 2026 (UTC)
:So maybe I previously misunderstood. Are you proposing separated pages for nominations (i.e. [[Wikiversity:Candidates for Curatorship]], [[Wikiversity:Candidates for Bureaucratship]], [[Wikiversity:Candidates for Custodianship]])? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:30, 5 May 2026 (UTC)
:: Yes. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:33, 5 May 2026 (UTC)

== Technical Request:  Courtesy link.. ==

[[Template_talk:Information#Background_must_have_color_defined_as_well]] [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
: I can't edit the template directly as it need an sysop/interface admin to do it. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
:: Also if the Template field of - https://en.wikiversity.org/wiki/Special:LintErrors/night-mode-unaware-background-color is examined, there is poential for an admin to clear a substantial proportion of these by implmenting a simmilar fix to the indciated templates (and underlying stylesheets). It would be nice to clear things like Project box and others, as many other templates (and thus pages depend on them.) :)
[[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)

:I think it would be best to grant you interface admin rights for a short period of time to make these changes. However, I still have doubts about the suitability of this solution, which may cause other problems and no one has explained to me why dark mode has to be implemented this way @[[User:ShakespeareFan00|ShakespeareFan00]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:43, 20 March 2026 (UTC)

: I would have reservations about holding such rights, which is why I was trying to do what I could without needing them. However if it is the only way to get the required changes made, I would suggest asking on Wikipedia to find technical editors, willing to undertake the changes needed. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:32, 21 March 2026 (UTC)

== WikiEducator has closed ==

Some of you may know of a similar project to Wikiversity, called [https://wikieducator.org/Main_Page WikiEducator], championed by [https://oerfoundation.org/about/staff/wayne-mackintosh/ Wayne Mackintosh][https://www.linkedin.com/posts/waynemackintosh_important-notice-about-the-oer-foundation-activity-7405113051688931329-Nhm9/][https://openeducation.nz/killed-not-starved/].

It seems [https://openeducation.nz/terminating-oer-foundation their foundation has closed] and they are no longer operating.

They had done quite a bit of outreach (e.g., in the Pacific and Africa) to get educators using wiki.

The WikiEducator content is still available in MediaWiki - and potentially could be imported to Wikiversity ([https://wikieducator.org/WikiEducator:Copyrights CC-BY-SA] is the default license).

The closing of WikiEducator arguably makes the nurturing of Wikiversity even more important.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:09, 1 April 2026 (UTC)

:I was never active there. If anyone has an account or is otherwise in contact, we may want to copy relevant information here or even at [[:outreach:]]. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:46, 1 April 2026 (UTC)

:: I reached out to [[User:Mackiwg~enwikiversity|Wayne]] in January, and he responded briefly but positively (while travelling).  I wrote to the low-traffic wikieducator mailing list today and got a nice [https://groups.google.com/g/wikieducator/c/r_yIyUw6ZIA reply] from [[user:SteveFoerster|Steve Foerster]] who's interested in helping. If we can figure out a migration path it would be great to adopt at least the main namespace pages here.  
:: A few questions that come to mind: 
:: - would people want to create matching user accounts
:: - are there any namespaces (user/talk?) that should not be moved over
:: We could look at how this was done for the [[m:Wikivoyage/Migration]] wikivoyage migration. &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 04:27, 1 May 2026 (UTC)
:::That's fantastic, SJ, that you've reached out and that Wayne, Steve, and Jim are receptive—and that you can help! -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:52, 1 May 2026 (UTC)

== Wikinews is ending ==

Apparently mainly due to low editorial activity, low public interest, but also failure to achieve the goals from the proposal for the creation of the project, the Wikinews project is ending after years of discussions ([[Meta:Proposal for Closing Wikinews|some reading]]).

And I would be interested to see how Wikiversity is doing in the monitored metrics. We probably have more editors than Wikinews had, but what about consumers and achieving the goals? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:14, 1 April 2026 (UTC)

:Wikiversity's biggest issue in recent times was the hosting of low-quality, trash content. Thankfully we've done a great job in removing pseudoscience and other embarrassingly trash content (Wikidebates, for example), but the biggest concern moving forward is proper maintenance IMO. I've caught several pseudoscience pages being created within the last few months that could easily have flown under the radar (ex, [[The Kelemen Dilemma: Causal Collapse and Axiomatic Instability]]), so I'd urge our custodians/curators to be on the lookout for this type of content. Usually an AI-overview can point this type of content out relatively well.
:In terms of visibility, I believe Wikiversity is a high-traffic project. I remember my [[Mathematical Properties]] showing up on the first page of Google when searching up &quot;math properties&quot; for the longest time (and is still showing up in the first page 'till this day!). Besides, Wikinews hosted a lot of short-term content (the nature of news articles), while Wikiversity hosts content that can still be useful a decade later (ex, [[A Reader's Guide to Annotation]]). 
:I think we are on a better path than we were a few months ago, and I do want to thank everyone here who has been helping out with maintaining our website! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:48, 1 April 2026 (UTC)
:For what it's worth, the group that did that study has since disbanded, so no one is monitoring the other sister projects in the same way. Additionally, Wikinews had some catastrophic server issues due to the maintenance of [[:m:Extension:DynamicPageList]] which don't apply here. Your questions are still worth addressing, but I just wanted to cut off any concern at the pass about Wikiversity being in the same precarious situation. Wikiversity is definitely the biggest &quot;lagging behind&quot; or &quot;failure&quot; project now that Wikinews is being shuttered, but I don't see any near- or medium-term pathway to closing Wikiversity. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:46, 2 April 2026 (UTC)

:[[w:en:Wikipedia:Wikipedia Signpost/2026-03-31/News and notes|Entirety of Wikinews to be shut down]] (Wikipedia Signpost) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:03, 11 April 2026 (UTC)

== Action Required: Update templates/modules for electoral maps (Migrating from P1846 to P14226) ==

Hello everyone,

This is a notice regarding an ongoing data migration on Wikidata that may affect your election-related templates and Lua modules (such as &lt;code&gt;Module:Itemgroup/list&lt;/code&gt;).

'''The Change:'''&lt;br /&gt;
Currently, many templates pull electoral maps from Wikidata using the property [[:d:Property:P1846|P1846]], combined with the qualifier [[:d:Property:P180|P180]]: [[:d:Q19571328|Q19571328]]. 

We are migrating this data (across roughly 4,000 items) to a newly created, dedicated property: '''[[:d:Property:P14226|P14226]]'''.

'''What You Need To Do:'''&lt;br /&gt;
To ensure your templates and infoboxes do not break or lose their maps, please update your local code to fetch data from [[:d:Property:P14226|P14226]] instead of the old [[:d:Property:P1846|P1846]] + [[:d:Property:P180|P180]] structure. A [[m:Wikidata/Property Migration: P1846 to P14226/List|list of pages]] was generated using Wikimedia Global Search.

'''Deadline:'''&lt;br /&gt;
We are temporarily retaining the old data on [[:d:Property:P1846|P1846]] to allow for a smooth transition. However, to complete the data cleanup on Wikidata, the old [[:d:Property:P1846|P1846]] statements will be removed after '''May 1, 2026'''. Please update your modules and templates before this date to prevent any disruption to your wiki's election articles.

Let us know if you have any questions or need assistance with the query logic. Thank you for your help! [[User:ZI Jony|ZI Jony]] using [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 17:11, 3 April 2026 (UTC)
&lt;!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&amp;oldid=29941252 --&gt;
:I didnt find such properties, so we are probably fine. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:00, 12 April 2026 (UTC)
:: +1 (agreed). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:19, 12 April 2026 (UTC)

== Enable the abuse filter block action? ==

In light of [[Special:AbuseLog/80178]] (coupon spam), I would like to propose enabling the block action for the abuse filter. Only custodians will be able to enable and disable that action on an abuse filter, and it is useful to block ongoing vandalism. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:12, 13 April 2026 (UTC)

:Seems like a good idea, almost all of the users which create such pages are spambots so this shouldn’t be a problem. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:41, 13 April 2026 (UTC)
:Can you explain some more (I am new to abuse filters)? It looks like the attempted edit was prevented? Which abuse filter?
:Note on your suggestion, have also reactivated Antispam Filter 12 - see [[WV:RCA]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:45, 15 April 2026 (UTC)
:: I am proposing that we activate the abuse filter block action, which if a user triggers an abuse filter, it would actually block the user in question - the same mechanism that a custodian would use to block users. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:11, 15 April 2026 (UTC)
:::OK, thankyou, that makes sense. And, reviewing the abuse filter 12 log, it would be helpful because it would prevent the need for manual blocking. But I don't see a setting for autoblocking? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:14, 15 April 2026 (UTC)
:::: I think it probably adds an autoblock. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:43, 16 April 2026 (UTC)
: [[User:Jtneill|Jtneill]] and [[User:PieWriter|PieWriter]], given that a little bit more than a week has passed and there is minimal consensus to activate the abuse filter block action, I filed [[phab:T424053]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:05, 21 April 2026 (UTC)
::Thank-you for doing this. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:03, 24 April 2026 (UTC)

== Advice needed: A Neurodiversity-inspired Idea/observation ==

If I want the greatest participation of others to &quot;provide constructive criticism to my idea&quot; or to &quot;shoot down my idea&quot; or &quot;idea&quot;.

What I've called it so far is &quot;The Neurodiversity-inspired Idea&quot;. At other times I used more sensationalist wording but here on Wikiversity I don't dare do that. I actually woke up with thinking about putting this into my userspace draft: &quot;Personal Observations Made By Meeting Autistic and Non-Autistic Adults&quot;.

My ultimate goal is to stop blathering about my &quot;idea&quot; to friend and family without feeling my &quot;methodology&quot; is going into any progressive direction whatsoever. My latest encounter was somewhat constructive though. A friend of a friend who worked with people presenting ideas in attempting to getting grants. I don't want a grant. I just want to figure out how I can express my &quot;idea&quot; in a way so that I can more clearly figure out what flaws it got.

At the same time I tend to overthink. If anyone thinks etherpad might be a good place and considering Wikimedia already got an etherpad at https://etherpad.wikimedia.org/ if anyone feels like they know me better in the future feel free to suggest a &quot;session&quot; on etherpad.

'''If I don't receive a reply to this in 1 week's time I will begin to explore this &quot;idea&quot; into my userspace''' unless you replied and refrained me from doing so, of course. Then maybe after &quot;developing it there&quot; I might reference it to you another future time here in the Colloquium, with my &quot;idea&quot; still in my userspace draft. This &quot;idea&quot; is sort of a burden, I'm happy I've made the choice to get rid of it and hopefully move on with my life, unless there is something to this &quot;idea&quot;. 

My failure is probably evident: I feel I haven't told you anything. Same happened to when I talked to friends and family. In danger of overthinking it further I'll publish this right now. I need to &quot;keep it together&quot; [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:36, 16 April 2026 (UTC)

:Good on you putting it out there ... and hitting publish :). I'd say go for it (no need to wait), give birth to your idea and share about it here and elsewhere. Let it take shape and see where it might go. In many ways, this is exactly what an open collaborative learning community should be doing. Others might not know well how to respond, so perhaps consider creating some questions to accompany the idea. Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:21, 16 April 2026 (UTC)
::Thank you for encouraging me in developing the idea.
::I have created a &quot;questions&quot; section in the draft which is visible in the table of contents now. My brain was &quot;frozen&quot; today metaphorically speaking in that I felt I had like a &quot;writer's block&quot; so the draft has more &quot;AI/LLM&quot; content than before. I used the LLM for generating questions. The answers are so far human-only.
::I've also created a subsection where I could add the prompts that made the LLM generate the questions. That could help people make better prompts perhaps. I've described what it is about inside of it and there are some chaotically written notes.
::[[Draft:The_Neurodiversity-inspired_Idea#Questions_that_might_encourage_the_development_of_this_idea_and_its_methodology]]
::My draft is missing stuff. Any questions that you contribute to my draft will probably help me and if I don't understand the questions I'll probably notify you and also at the same time &quot;feed them&quot; to an LLM and ask in my input like &quot;explain in simple words what this question means, what is it searching for?&quot; etc. while I wait for an answer. If you have any more feedback please give it to me here or on the Draft page, its talk page or my user talk page. Thank you for helping me! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 21:20, 18 April 2026 (UTC)
::Today I woke up with not only thinking about supplying questions along with the &quot;idea&quot; but also answers. ie. Is it possible to &quot;test&quot; this idea? Is it possible to create one or multiple hypotheses based on this &quot;idea&quot;?(etc.) I've thought about this before in this &quot;idea&quot; but since I'm beginning to add to Wikiversity what was previously 'locked in my mind' it's also easier for me to see what I've done so far. Thank you for this comment! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 09:11, 23 April 2026 (UTC)
:May I think that you should not add deadlines ; being read, and rising interest for collaboration, or even simply for exchange of thoughts, such an effective meeting event loads a huge bunch of unprobability, which time can help to… somehow diminish. Maybe, I would advice you having a central place for developping your ideas, your needs, your advances, maybe a page in your own user zone, and from time to time, depending your feeling, it could be every trimester or so, or more frequently, you could write a short account of progress (or even of no progress), or a call for participation, in such a place as this present one ; I think that will increase much exposure of your projet. Maybe also, if you can find a project name, not necessarily very meaningfull by itseilf (at least it will gain signification with time, as your project develops), that will serve as a kind-of hook, and make your announcement titles more visible. Best regards (and my excuses for my poor command of English, which seems to be unplease an anti-abuse filter, &quot;Questionable Language (profanity)&quot;, which I don't understand…). My few cents. --  [[User:Eric.LEWIN|Eric.LEWIN]] ([[User talk:Eric.LEWIN|discussion]] • [[Special:Contributions/Eric.LEWIN|contributions]]) 10:06, 17 April 2026 (UTC)
::Sorry about the false positive on the profanity filter - I've fixed it. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:26, 17 April 2026 (UTC)
:::&quot;May I think that you should not add deadlines ; being read, and rising interest for collaboration, or even simply for exchange of thoughts, such an effective meeting event loads a huge bunch of unprobability, which time can help to… somehow diminish.&quot;
::Thank you Eric for this comment. Trust in time is how I interpret it. I should not feel like I need to be in a hurry. I'll try to give this time. Thank you!
:::&quot;Maybe, I would advice you having a central place for developping your ideas, your needs, your advances, maybe a page in your own user zone, and from time to time, depending your feeling, it could be every trimester or so, or more frequently, you could write a short account of progress (or even of no progress), or a call for participation, in such a place as this present one ; I think that will increase much exposure of your projet.&quot;
::A central place for developing or making &quot;project notes&quot; regarding the Neurodiversity idea on my userspace, I might need that, like a diary or &quot;project notes&quot; of the Neurodiversity idea similar to my course notes regarding my experience with Coursera.
::Any actions I take are going to be related to my Userspace from now on but I'll also update the draft when necessary. Now in the beginning I might be working daily to once every 3 days on both the draft and the daily notes I plan to make.
:::&quot;Maybe also, if you can find a project name, not necessarily very meaningfull by itseilf (at least it will gain signification with time, as your project develops), that will serve as a kind-of hook, and make your announcement titles more visible.&quot;
::Thank you for the advice. I was brainstorming yesterday about it. I concluded that since I've not yet developed a methodology that adheres to &quot;Do no harm&quot; and this is my first time working my &quot;idea&quot; into a way that is compatible with how projects develop on English Wikiversity this is new to me. My methodology isn't developed and therefore trying to get attention to my project through a name can wait. Yesterday I figured out a silly title that has nothing to do with the project: &quot;Planetary Awareness Potato Cabbage Rolls&quot; or something like that. Google output read that no such thing exists so I wanted it mainly to be unique. I don't want to raise attention that I'm unsure whether I'll actually be capable of developing a methodology for but project notes is my best bet so far in tracking my progress. Every day I think about this &quot;idea&quot; but I need to improve the important parts.
:::&quot;Best regards (and my excuses for my poor command of English, which seems to be unplease an anti-abuse filter, &quot;Questionable Language (profanity)&quot;, which I don't understand…). My few cents.&quot;
::You added great points and I felt that I was helped by you! I encourage you to post again and I can understand that interacting with any kind of automated filter can be discouraging and can be for me too! Thank you for giving me feedback! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 16:01, 18 April 2026 (UTC)

== Add some user rights to the curator user group? ==

By default, only custodians have the ability to mark new pages as patrolled (&lt;code&gt;patrol&lt;/code&gt;) and have their own page creations automatically marked as patrolled (&lt;code&gt;autopatrol&lt;/code&gt;). I am proposing both of the following:
* Curators can mark new pages as patrolled, helping on reducing the backlog of new, unpatrolled pages.
* New pages made by curators will be automatically marked as patrolled by the MediaWiki software.
Before we implement this, I would suggest implementing a proposed guideline for marking new pages as patrolled for curators and custodians.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:32, 17 April 2026 (UTC)

:Agree, &lt;s&gt;also can we also allow curators to undelete pages since they already have the rights to delete them?&lt;/s&gt; [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:54, 18 April 2026 (UTC)
::I think the requirement that undelete NOT be included came from above (meta / stewards / central office). Having access to the undelete page gives access to information that is restricted by their policies to admins (custodians and bureaucrats). -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:12, 18 April 2026 (UTC)
::: [[User:PieWriter|PieWriter]], unless if requests for curator and custodian should be RfA-like processes (that is, including voting and comments), then I have to agree with Dave above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:03, 18 April 2026 (UTC)
::::Oh, I didn’t realise that. Withdrawing my comment.. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:08, 19 April 2026 (UTC)
:{{support}} Seems reasonable and would reduce overhead. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 14:35, 18 April 2026 (UTC)
:'''Agree''', implement it also to [[Wikiversity:Curators]] proposal please. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:11, 18 April 2026 (UTC)
: I went ahead and filed [[phab:T424445]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:39, 26 April 2026 (UTC)

== [[Wikiversity:Curators|Curators and curators policy]] ==

How does it come, that Wikiversity has curators, but Curators policy is still being proposed? How do the curators exists and act if the policy about them havent been approved yet? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:33, 16 October 2025 (UTC)

:It looks as if it is not just curators. The policy on Bureaucratship is still being proposed as well. See [[Wikiversity:Bureaucratship]]. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 18:33, 27 October 2025 (UTC)

:I think its just the nature of a small WMF sister project in that there are lots of drafts, gaps, and potential improvements. In this case, these community would need to vote on those proposed Wikiversity staff policies if we think they're ready. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:08, 3 December 2025 (UTC)
:What? I thought you were getting it approved, Juandev... :) [[User:I&amp;#39;m Mr. Chris|I&amp;#39;m Mr. Chris]] ([[User talk:I&amp;#39;m Mr. Chris|discuss]] • [[Special:Contributions/I&amp;#39;m Mr. Chris|contribs]]) 14:20, 12 February 2026 (UTC)
::Yeah I think this one is important too and we need to aprove it too @[[User:I'm Mr. Chris|I'm Mr. Chris]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 12 February 2026 (UTC)
:::I thinks its ready to made into a policy, it seems to be complete and informative about what the rights does and how to get it. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:08, 15 February 2026 (UTC)
::::Agree -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:00, 27 March 2026 (UTC)

Let's make this the official discussion about adopting the [[Wikiversity:Curators|curators policy]] policy. Your comments are invited and welcome. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:40, 24 April 2026 (UTC)
: There were two similar Colloquium threads in separate places about the proposed curators policy. So I've moved them to be adjacent. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:42, 1 May 2026 (UTC)

== Wikiversity:Curators to become a policy ==

I've looked at the discussions about the Curators policy, I've looked at the practices, and it seems to me that there is no dispute about the wording of the policy, and what's more, the community has been using this proposal as if it were an offical policy for several years. Therefore, I propose that [[Wikiversity:Curators]] become a policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:35, 18 April 2026 (UTC)

:{{support}} —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 18 April 2026 (UTC)
:{{support}} —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:21, 18 April 2026 (UTC)
: {{support|Yes, please}}. Especially after when I and PieWriter proposed above, I agree. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:27, 24 April 2026 (UTC)
:: @[[User:Juandev|Juandev]]; as of now, curators now have the user rights &lt;code&gt;autopatrol&lt;/code&gt; and &lt;code&gt;patrol&lt;/code&gt;. Perhaps we should also include that in [[Wikiversity:Custodianship]]? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 12:07, 30 April 2026 (UTC)
:::You meant [[Wikiversity:Curators]] @[[User:Codename Noreste|Codename Noreste]]? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:15, 5 May 2026 (UTC)
:::: I agree that we must develop what rules curators should follow when marking new pages as patrolled; the same can be added for custodians since they can also mark new pages as patrolled. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:37, 5 May 2026 (UTC)
:{{support}} -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:42, 1 May 2026 (UTC)
:{{Support}} per nom. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 13:32, 1 May 2026 (UTC)

== Inactive curators ==

Hello, even though [[Wikiversity:Curators]] is not a policy yet, there are curators listed here that have been inactive for two years or more:
* {{user|Cody naccarato}} (last edit on 13 Dec 2022, last logged action on 10 Dec 2022)
* {{user|Praxidicae}} (last edit on 10 Sep 2022, last logged action on 12 Sep 2022)
[[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 19 April 2026 (UTC)

:Yup, I would remove the rights. To get the rights back if theyll come back should not be a big deal. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:08, 24 April 2026 (UTC)
:: When they don't reply by May 19, feel free (or any custodian) to do so. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:28, 25 April 2026 (UTC)

== Is anyone interested in Neurodiversity? ==

Is anyone interested in Neurodiversity? Is there anyone here who is interested for Neurodiversity to be &quot;something more&quot; than it already is? Does anyone here consider Neurodiversity one of the &quot;harder topics&quot; to work on or discuss? Does anyone here have an opinion about the [[Neurodiversity Movement]]? So these questions don't appear like &quot;out of a vacuum&quot; I can tell you a bit about my background:

Many years ago I got a psychiatric diagnosis &quot;Asperger's&quot;. After I stepped out of the office and my Äsperger's was 'concluded', I stepped out into the street and thought my first negative thought(but the positive thought followed after). The thought was about concentration camps in the second world war and that the world seemed to be going into the direction of &quot;labeling others&quot;. I was unsure whether this was &quot;real science&quot; and sort of &quot;challenged myself&quot; to make up my own mind after meeting people that had been given this diagnosis. The more adults with this diagnosis I met the more I started seeing &quot;patterns&quot;.

Was it a coincidence that the first person with Asperger's I met reminded me about my father later after I had plenty of times of experience with interacting with him? None of the people I interacted with online through IRC text chat...I felt I got any clue about how &quot;their brains work&quot;. Only when I met one person from the Asperger's chat community in person we both realized that whatever we experienced was akin to the &quot;chaos theory&quot;. He told me about &quot;chaos theory&quot; while I didn't know even what that term meant but I guess I 'read between the lines'. My question that I linger on still today is &quot;did he understand about me what I think I understood about him?&quot;? That our brains had the same configuration? Most autistic adults who meet other autistic adults usually get disappointed. They think the diagnosis will help them meet somebody like themselves and then they realize the great diversity in the autistic spectrum created by Psychiatry.

I later stopped interacting with autistic communities that much, I felt that it did not benefit me. Also Neurodiversity's &quot;neurotypes&quot; interested me for a while until I realized I had &quot;misunderstood everything&quot; about them and how they are used in the Neurodiversity Movement or &quot;Neurodiversity community&quot; if that even can precisely be defined? I doubt it but if you want to contribute to the [[Neurodiversity Movement]]. My previous attempts failed as I got more and more confused. I think a community project needs a community. With a lack of that I don't think it is worth my time. If any of you would like to work on that project let me know on my talk page.

So I was kinda lost and was talking to my friend and psychologist and I realized if I never talk about my idea to anyone in a &quot;comprehensive way&quot; or show that it matters to me nothing is going to ever happen. So I started talking about my &quot;idea&quot; more. Nobody could understand the &quot;idea&quot; because I had not developed my skills regarding where to start...although the process had already started &quot;automatically&quot; and that's why I often think of &quot;well my brain sort of activated me&quot;. I don't feel like I did have a plan and this idea happened. It happened &quot;by itself&quot;. My brain reacted to what I was seeing in a video or stream.

I value interaction highly in this idea. I think it would be helpful to make a community of people who are not paranoid about stuff that can express itself like &quot;don't analyze me!&quot;, &quot;don't compare me to anyone!&quot;. 

On the contrary, more often than not those adults who were diagnosed were actually openly comparing themselves with each other and I think that is healthy in a &quot;science&quot; way if done the &quot;right way&quot; which probably means &quot;Do no harm&quot;.

I found video material is important but I'm very unsure if uploading own video material to Wikimedia Commons would constitute a &quot;reasonable&quot; use of the resources there. Maybe somebody here needs to ask more questions to me that I should answer before that happens. I also know the '''be bold''' so I could just do what I think might be ok. Though I work better in a group as long as I know what &quot;group configurations&quot; help me. This is in a non-profit way. Since the state supported me this might be a way I am trying to &quot;give back&quot; to the state and &quot;the world&quot;. May seem overly ambitious and crazy but this thing gives me energy. It gives me hope when trying to develop this idea. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:47, 23 April 2026 (UTC)

:Thanks for sharing. There is plenty of room for neurodiversity community learning. However, the challenge I think is that the intersection of those interested in (a) ND, and (b) English Wikiversity might be very small (e.g., 1!) at this point in time.
:But don't give up hope. For example, Wikipedia has many more ND-interested editors; maybe consider reaching out to see who might be interested:
:[[w:Category:Wikipedians interested in neurodiversity]]
:You could also start an equivalent category here:
:[[:Category:Wikiversitarians interested in neurodiversity]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:46, 6 May 2026 (UTC)

== Request for comment (global AI policy) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;A [[:m:Requests for comment/Artificial intelligence policy|request for comment]] is currently being held to decide on a global AI policy. {{int:Feedback-thanks-title}} [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 00:58, 26 April 2026 (UTC)&lt;/bdi&gt;
&lt;!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30424282 --&gt;

== Coming over From wikinews ==

Any chance someone could help me if you are allowed to write news articles here since wikinews is going read only mode soon, thank you! [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:43, 1 May 2026 (UTC)

:The scope of Wikiversity is very broad and is basically about more-or-less any learning material. We have made it a point to not have duplicative content of other WMF projects, but since Wikinews is being shuttered, I personally am fine with writing news articles here. One thing that is not controversial at all is a learning resource &lt;em&gt;about&lt;/em&gt; how to write news: that could be hugely useful here and could involve the process of writing news stories to learn and to share back and forth with an editor or fact-checker. In fact, I'd support an entire namespace dedicated to keeping the notion of Wikinews alive here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:38, 1 May 2026 (UTC)
::Thank you so much! How do I start? Cheers! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:07, 2 May 2026 (UTC)
:::I think it's premature to start just making news articles en masse, but if you want to start discussing the topic of citizen journalism, you can do that now. [[:Category:Journalism]] already has some material, so you can start by seeing what we already have, how you can refine that, etc. You can definitely have learning resources with collaborators who want to learn about journalism ASAP. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:24, 2 May 2026 (UTC)
::::thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:38, 2 May 2026 (UTC)
::::If I could try and start one News Article could you please tell me how to go about it? Like what style of writing like Wikinews or something else? Thank you Justin! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:48, 2 May 2026 (UTC)
:::::Honestly, there are very few policies and guidelines here. I think the best way to write a news story would be in a manner that is obvious and instructive. So, for instance, it's common to use the &quot;pyramid style&quot; when you're writing news, so if you were to write a story that makes it very clear that you are using that approach, that would be helpful. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:08, 2 May 2026 (UTC)
::::::cool thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 02:13, 2 May 2026 (UTC)

== Language learning  ==

toki! I am trying to add or see what the toki pona language learning stuff on here is but I don't see anything that is language learning for anything. [[User:Jan Imon|Jan Imon]] ([[User talk:Jan Imon|discuss]] • [[Special:Contributions/Jan Imon|contribs]]) 23:13, 2 May 2026 (UTC) —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:29, 3 May 2026 (UTC)

:We have language materials ([[:Category:Languages]], [[World Languages]], [[Portal:Foreign Language Learning]], [[Portal:Multilingual Studies]]). They are not as developed as I think we would all like and there's not any coverage of Toki Pona, but in principle, we could and would like that. You can also see [[:b:Subject:Languages]] at our sister project Wikibooks. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:33, 3 May 2026 (UTC)

== Timeline format? ==

I’ve been working on the World War II articles, including the [[World War II/Timeline|timeline]], and is there a specific timeline format that should be used? Right now it’s just a table, and there’s no separation between different periods/phases of the war.

I don’t want to use [[mw:Extension:EasyTimeline]] because this will be displaying dates and not time periods. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 01:35, 4 May 2026 (UTC)

:I dont think we have a policy or guideline, how to format a timeline. But you may try to browes wikiversity by Google if someone was dealing with this in the past somewhow @[[User:PhilDaBirdMan|PhilDaBirdMan]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:23, 5 May 2026 (UTC)
::+1 - there's no specific guideline on how to format a timeline, it's really up to you. In my opinion I think the timeline is good. I'd personally bold the dates just to make it easier to separate it from the event description, but that's my personal 2 cents. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:18, 5 May 2026 (UTC)
:::I’ll probably remove links to the dates/years, they’re just Wikipedia pages that shouldn’t be over linked to. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 00:39, 6 May 2026 (UTC)

== Interface administrator for Codename Noreste ==

Hello, everyone. I am requesting interface administrator access on this wiki.

The main reasoning is that I would benefit from having the user right &lt;code&gt;editinterface&lt;/code&gt;, which would allow me to make dark mode changes to pages in the MediaWiki namespace, add &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-parser-output&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt; to some interface pages using templates, handle interface-protected edit requests, and similar stuff. Additionally, I have some knowledge of CSS, and I would like to assist with modifying CSS pages whenever necessary, such as moving MediaWiki common.css code to TemplateStyles CSS pages.

I am requesting the maximum time that is allowed per the [[Wikiversity:Interface administrators|policy]], and I have 2FA enabled on my account. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:55, 6 May 2026 (UTC)

*{{support}} Globally trusted user. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:07, 6 May 2026 (UTC)
*{{support}} Trusted and knowledgeable. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:35, 6 May 2026 (UTC)
*{{support}} WV would benefit from this. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 08:32, 6 May 2026 (UTC)</text>
      <sha1>kpsfukxajaj8gpft4lt37bcjipwful2</sha1>
    </revision>
    <revision>
      <id>2807862</id>
      <parentid>2807809</parentid>
      <timestamp>2026-05-07T09:14:07Z</timestamp>
      <contributor>
        <username>Juandev</username>
        <id>2651</id>
      </contributor>
      <comment>/* Interface administrator for Codename Noreste */ +1</comment>
      <origin>2807862</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="76195" sha1="8d3tjmarm6y8zng6chilv38dbziln5l" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;
== Requested update to [[Wikiversity:Interface administrators]] ==

Currently, [[Wikiversity:Interface administrators]] is a policy that includes a caveat that interface admins are not required long-term and that user right can only be added for a period of up to two weeks. I am proposing that we remove this qualification and allow for indefinite interface admin status. I think this is useful because there are reasons for tweaking the site CSS or JavaScript (e.g. to comply with dark mode), add gadgets (e.g. importing Cat-a-Lot, which I would like to do), or otherwise modifying the site that could plausibly come up on an irregular basis and requiring the overhead of a bureaucrat to add the user rights is inefficient. In particular, I am also going to request this right if the community accepts indefinite interface admins. Thoughts? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:23, 17 August 2025 (UTC)

:And who will then monitor them to make sure they don't damage the project in any way, or abuse the rights acquired in this way? For large projects, this might not be a problem, but for smaller projects like the English Wikiversity, I'm not sure if there are enough users who would say, something is happening here that shouldn't be happening. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:28, 20 August 2025 (UTC)
::Anyone would be who. This argument applies to any person with any advanced rights here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 10:46, 20 August 2025 (UTC)
:I think it is reasonable to allow for longer periods of access than 2 weeks to interface admin and support adjusting the policy to allow for this flexibility. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:57, 2 December 2025 (UTC)
::+1 —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:38, 25 January 2026 (UTC)
:@[[User:Koavf|Koavf]] I agree that the two-week requirement could be revised, but wouldn’t people just request access for a specific purpose anyway? Instead of granting indefinite access, they should request the specific time frame they need the rights for—until the planned fixes are completed—and then request an extension if more time is required. We could remove the two-week criterion while still keeping the access explicitly temporary. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:48, 25 January 2026 (UTC)
::I just don't see why this wiki needs to be different than all of the others. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:18, 25 January 2026 (UTC)
:::There isn’t really much of a need for a permanent one at this point in time [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:53, 25 January 2026 (UTC)
:I quite agree with this proposal, so long as they perform the suggested changes as mentioned here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 04:06, 26 January 2026 (UTC)
:: Just to clarify, I support '''indefinite interface admin status'''. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:34, 13 April 2026 (UTC)
:I think there is decent consensus for lengthening this, but not necessarily for indefinite permissions, so does anyone object to me revising it to the standard being 120 days instead of two weeks? I'll check back on this thread in three weeks and if there's no objection, I'll make the change. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:47, 13 April 2026 (UTC)
::Sure [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:27, 13 April 2026 (UTC)
::Thanks for proposing this, Justin. I agree with the proposal to lengthen the interface admin period from 2 weeks but not indefinitely. Can I check the source(s) for the standard being 120 days (I'm guessing policies on other projects or maybe global policy?)? In any case, I think it is reasonable for us to adopt a similar period. However, note on the current policy discussion page notes from @[[User:Dave Braunschweig|Dave Braunschweig]] arguing for shorter periods to lower risk, that's why it is 2 weeks. But if there are projects that need longer access, that should also be accommodated. Maybe we could adjust the policy to specify that ''interface admin rights can be given for 14 to 120 days depending on how long is required and what is supported by the community''. Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:29, 24 April 2026 (UTC)
:::There was there was no source for 120: it was just more than 14 and less than infinity. The &quot;14 to 120&quot; also seems reasonable. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 14:33, 24 April 2026 (UTC)
::: On some small/medium-sized wikis, such as English Wikibooks and English Wikiquote for example, indefinite interface administrator access for administrators is allowed, but they tend not to make changes to the CSS and JS page changes unless it's truly necessary. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:34, 24 April 2026 (UTC)
:::It's a good idea to make the length of this right on request or allow to be prolonged. However, IA should test large changes somewhere else, for example on the en.wv mirror, and only after testing it on the mirror, adapt it to the live version. That means I can't imagine a time-consuming operation right now. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:04, 24 April 2026 (UTC)
::::Sorry, what mirror is this? Are you talking about beta.wv? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:32, 24 April 2026 (UTC)
:::::Not beta.wv. Basically somewhere else then on a live wiki. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:59, 24 April 2026 (UTC)
:::::: Wouldn't testing on a user's own common.css page work anyway? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:36, 24 April 2026 (UTC)
:Change made here: https://en.wikiversity.org/w/index.php?title=Wikiversity%3AInterface_administrators&amp;diff=2807543&amp;oldid=2806289 —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:35, 4 May 2026 (UTC)

== [[Template:AI-generated]] ==

After going through the plethora of ChatGPT-generated pages made by [[User:Lbeaumont|Lbeaumont]] (with many more pages to go), I'd like community input on this proposal to [[Wikiversity:Artificial intelligence]] that I think would be benefical for the community:
*Resources generated by AI '''must''' be indicated as so through the project box, [[Template:AI-generated]], on either the page or the main resource (if the page is a part of a project).
I do not believe including a small note/reference that a page is AI-generated is sufficient, and I take my thinking from [[WV:Original research|Wikiversity's OR policy]] for OR work: ''Within Wikiversity, all original research should be clearly identified as such''. I believe resources created from AI should also be clearly indicated as such, especially since we are working on whether or not AI-generated resources should be allowed on the website (discussion is [[Wikiversity talk:Artificial intelligence|here]], for reference). This makes it easier for organizational purposes, and in the event ''if'' we ban AI-generated work. 

I've left a message on Lee's talk page over a week ago and did not get a response or acknowledgement, so I'd like for the community's input for this inclusion to the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:53, 26 January 2026 (UTC)

:I believe that existing Wikiversity policies are sufficient. Authors are responsible for the accuracy and usefulness of the content that is published. This policy covers AI-generated content that is: 1) carefully reviewed by the author publishing it, and 2) the source is noted.   [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 19:38, 27 January 2026 (UTC)
::A small reference for pages that are substantially filled with Chat-GPT entries, like [[Real Good Religion]], [[Attributing Blame]], [[Fostering Curiosity]], are not sufficient IMO and a project box would be the best indicator that a page is AI-generated (especially when there is a mixture of human created content AND AI-generated content, as present in a lot of your pages). This is useful, especially considering the notable issues with AI (including hallucinations and fabrication of details), so viewers and support staff are aware. These small notes left on the pages are not as easily viewable as a project box or banner would be. I really don't see the issue with a clear-label guideline. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:34, 27 January 2026 (UTC)
::{{ping|Lbeaumont}} I noticed your reversions [https://en.wikiversity.org/w/index.php?title=Exploring_Existential_Concerns&amp;diff=prev&amp;oldid=2788278 here] &amp; [https://en.wikiversity.org/w/index.php?title=Subjective_Awareness&amp;diff=prev&amp;oldid=2788257 here]. I'd prefer to have a clean conversation regarding this proposition. Please voice your concerns here. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:53, 28 January 2026 (UTC)
:::Regarding Subjective Awareness, I distinctly recall the effort I went to to write that the old-fashioned way. It is true that ChatGPT assisted me in augmenting the list of words suggested as candidate subjective states. This is a small section of the course, is clearly marked, and makes no factual claim. Marking the entire course as AI-generated is misleading. I would have made these comments when I reverted your edit; however, the revert button does not provide that opportunity.
:::Regarding the Exploring Existential Concerns course, please note this was adapted from my EmotionalCompetency.com website, which predates the availability of LLMs. The course does include two links, clearly labeled as ChatGPT-generated. Again, marking the entire course as AI-generated is misleading.
:::On a broader issue, I don't consider your opinions to have established a carefully debated and adopted Wikiversity policy. You went ahead and modified many of my courses over my clearly stated objections. Please let this issue play out more completely before editing my courses further. Thanks.   [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 15:11, 29 January 2026 (UTC)
::::Understood, and I respect your position. I apologize if my edits were seen as overarching. We could change the project box to &quot;a portion of this resource was generated by AI&quot;, or something along those lines. Feel free to revert my changes where you see fit, and I encourage more users to provide their input. EDIT: I've made changes to the template to indicate that a portion of the content has been generated from an LLM. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:50, 29 January 2026 (UTC)
:::::Thanks for this reply. The new banner is unduly large and alarming. There is no need for alarm here. The use of AI is not harmful per se. Like any technology, it can be used to help or to harm. I take care to craft prompts carefully, point the LMM to reliable source materials, and to carefully read and verify the generated text before I publish it. This is all in keeping with long-established Wikiversity policy. We don't want to use a  [[w:One-drop_rule|one-drop rule]] here or cause a [[w:Satanic_panic|satanic panic]]. We can learn our lessons from history here. I don't see any pedagogical reason for establishing a classification of &quot;AI generated&quot;, but if there is a consensus that it is needed, perhaps it can be handled as just another category that learning resources can be assigned to. I would rather focus on identifying any errors in factual claims than on casting pejorative bias toward AI-generated content. An essay on the best practices for using LMM on Wikiveristy would be welcome.   [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 15:58, 30 January 2026 (UTC)
::::::The new banner mimics the banner that is available on the English Wikibooks (see [[b:Template:AI-generated]] &amp; [[b:Template:Uses AI]]), so my revisions aren't unique in this aspect. At this point, I'd welcome other peoples' inputs. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:40, 30 January 2026 (UTC)

== How do I start making pages? ==
Is there a notability guideline for Wikiversity? What is the sourcing policy for information? What is the Manual of Style? What kind of educational content qualifies for Wikiversity? All the introduction pages are a bit unclear. 
[[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 02:25, 28 January 2026 (UTC)

:{{ping|VidanaliK}} Welcome to Wikiversity! I've left you a welcome message on your talk page. That should help you out. Make sure to especially look at [[Wikiversity:Introduction]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 03:11, 28 January 2026 (UTC)
::It says that I can't post more pages because I have apparently exceeded the new page limit. How long does it take before that new page limit expires? [[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 16:57, 28 January 2026 (UTC)
:::This is a restriction for new users so that Wikiversity is not hit with massive spam. As for when this limit will expire, it should be a few days or after a certain number of edits. It's easy to overcome, though I do not have the exact numbers atm. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:08, 29 January 2026 (UTC)
::::OK, I think I got past the limit. [[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 17:21, 29 January 2026 (UTC)
==Why does it feel like Wikiversity is no longer really active anymore?==
I've been looking at recent changes, and both today and yesterday there haven't been many changes that I haven't made; it feels like walking through a ghost town, is this just me or is Wikiversity not really active anymore? [[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 03:54, 30 January 2026 (UTC)

:There is fewer people editing these days compared to the past. Many newcomers tend to edit in Wikipedia instead. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 06:39, 30 January 2026 (UTC)
:It’s a little slow, but I’m happy to know that Wikiversity is a place that I think should provide value even if the activity of editors fluctuates. If it’s any consolation your edits may be encouraging for some anonymous newcomer to start edits on their own! I think it’s hard to build community when there is such a wide variety of interests and a smaller starting userbase. Also sometimes the getting into a particular topic that already exists can be intimidating because some relics (large portals, school, categories, etc.) have intricate, unique and generally messy levels of organization. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 22:16, 9 March 2026 (UTC)
:I'd say it comes down to working hard for Wikiversity, basically if somebody or a group of people will start presenting good ideas and they turn out to be provably stable.
:I even asked Google's &quot;AI Mode&quot;, what is Wikiversity famous for? Unfortunately it could not answer that.
:Simply, we have not made Wikiversity famous by presenting really provable stable ideas yet. My hope is that this time might come. Perhaps even this year 2026!
:Hope dies last. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:12, 27 April 2026 (UTC)

== Inactivity policy for Curators ==

I was wondering if there is a specific inactivity polity for curators (semi-admins) as I am pretty sure the global policy does not apply to them as they are not ''fully'' sysops. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:20, 15 February 2026 (UTC)

:Unfortunately, I don't see an inactivity policy, but if we were to create such a new policy for curators, it should be the same for custodians (administrators). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:45, 15 February 2026 (UTC)
::@[[User:Codename Noreste|Codename Noreste]] There is currently none, that I could find, for custodians either. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:47, 17 February 2026 (UTC)
:::I think we should propose a local inactivity policy for custodians (and by extension, curators), which should be at least one year without any edits ''and'' logged actions. However, I don't know which page should it be when the inactivity removal procedure starts. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:53, 17 February 2026 (UTC)
::::@[[User:Codename Noreste|Codename Noreste]] In theory, there should be a section added at [[WV:Candidates for custodianship]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:55, 17 February 2026 (UTC)
::::: To be consistent with the [[meta:Admin activity review|global period of 2 years inactivity]] for en.wv  [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship#How are bureaucrats removed?|Bureaucrats]] we could add something like this to [[Wikiversity:Curators]]:
::::::The maximum time period of inactivity &lt;u&gt;without community review&lt;/u&gt; for curators is two years (consistent with  the [[:meta:Category:Global policies|global policy]] described at [[meta:Admin activity review|Admin activity review]] which applies for [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship|Bureaucrats]]). After that time a custodian will remove the rights.
::::: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:51, 27 March 2026 (UTC)
:::::Yup, I agree with Jtneill, there is a policy proposal for Wikiversity:Curators, where it should be logically deployed. The question is if we are ready to aprove the policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:43, 17 April 2026 (UTC)
:::::: I agree, but we should notify the colloquium about inactive curators, just like a steward would do for inactive custodians and bureaucrats per [[:m:Admin activity review|AAR]]. What is the minimum timeframe an inactive curator should receive so they can respond they would keep their rights? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:49, 17 April 2026 (UTC)
:I incorporated these suggestions into the proposed curators policy. Please review/comment/improve. Summary: 2 years, notify curator's user page, then remove rights after 1 month: [[Wikiversity:Curators#Inactivity]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:59, 24 April 2026 (UTC)
:: @[[User:Jtneill|Jtneill]] I created [[Template:Inactive curator]] for this. Feel free to make any changes or improvements. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:29, 24 April 2026 (UTC)

== [[Wikiversity:Artificial intelligence]] to become an official policy ==
{{Archive top|After running for a week, there is consensus, alongside comments, for [[Wikiversity:Artificial intelligence]] to be implemented as an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:27, 17 April 2026 (UTC)}}
With the introduction of AI-material, and some material just plain disruptive, its imperative that Wikiversity catches up with its sister projects and implements an official AI policy that we can work with. The recent issue of [[User:Lbeaumont|Lbeaumont]]'s 50+ articles that contain significantly large AI-generated material has made me came to the Colloquium. This user has also been removing the [[Template:AI-generated]] template from their pages, calling it &quot;misleading&quot;, &quot;alarmist&quot;, and &quot;pejorative&quot; - which is all just simply nonsensical rationales. Not to even mention this user's contributions to the English Wikipedia have been [https://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Inner_Development_Goals contested] and [https://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Multipolar_trap removed] a couple of times (for being low-quality and clearly LLM-generated), highlighting the need for an actual policy to be implemented here on Wikiversity. I would like to ping {{ping|Juandev}} and {{ping|Jtneill}} for their thoughts as well, since I'd like this to be implemented as soon as possible. 

Wikiversity has a significant issue with implementing anti-disruptive measures, hence why we have received numerous complaints as a community about our quality. I originally was reverting the removal of the templates, but realized that this is still a proposed policy, which it shouldn't be anymore. It should be a recognized Wikiversity policy. 14:54, 10 March 2026 (UTC) —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:54, 10 March 2026 (UTC)

:@[[User:Atcovi|Atcovi]] '''I agree''' that the draft, should become official policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:00, 10 March 2026 (UTC)
:I provided a detailed response at: [[Wikiversity talk:Artificial intelligence#Evolving a Wikiversity policy on AI]]
:I will appreaciate it if you consder that carefully. [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 22:49, 10 March 2026 (UTC)
:Agree it should become official Wikiversity policy on the condition &lt;u&gt;that point point 5 is about [significant/substantial] LLM-generated text specifically&lt;/u&gt;. Not a good idea to overuse it, it should be added when there is substantial AI-generated text on the page, not for other cases. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:37, 11 March 2026 (UTC)
:What policy is being debated? Is it the text on this page, which is pointed to by the general banner, or the text at:   [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence,]]   which is pointed to by the specific banner? Let's begin with coherence on the text being debated. Thanks!  [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 11:49, 17 March 2026 (UTC)
::@[[User:Lbeaumont|Lbeaumont]] This is a call for approval of the new Wikiversity policy. You expressed your opinion [[Wikiversity talk:Artificial intelligence#Evolving a Wikiversity policy on AI|on the talk page of the proposal]], I replied to you and await your response.When creating policies, it is necessary to propose specific solutions. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:12, 17 March 2026 (UTC)
:::Toward a Justified and Parsimonious AI Policy 
:::As we collaborate to develop a consensus policy on the use of Large Language Models, it is wise to begin by considering the needs of the various stakeholders to the policy.
:::The stakeholders are:
:::1)     The users, 
:::2)     The source providers, and
:::3)     The editors
:::There may also be others with a minor stake in this policy, including the population at large.
:::The many needs of the users are currently addressed by long-standing [[Wikiversity:Policies|Wikiversity policies]], so we can focus on what, if any, additional needs arise as LLMs are deployed. 
:::As always, users need assurance that propositional statements are accurate. This is covered by the existing policy on [[Wikiversity:Verifiability|verifiably]]. In addition, it is expected by both the users and those that provide materials used as sources for the text are [[Wikiversity:Cite sources|accurately attributed]]. This is also covered by [[Wikiversity:Cite sources|existing policies]].
:::To respect the time and effort of editors, a parsimonious policy will unburden editors from costly requirements that exceed benefits to the users.
:::Finally, it is important to recognize that because attention is our most valuable seizing attention unnecessarily is a form of theft. 
:::The following proposed policy statement results from these considerations:
:::Recommended Policy statement:
:::·       Editors [[Wikiversity:Verifiability|verify the accuracy]] of propositional statements, regardless of the source.
:::·       Editors [[Wikiversity:Cite sources|attribute the source]] of propositional statements. In the case of LLM, cite the LLM model and the prompt used.
:::·       Use of various available templates to mark the use of LLM are optional. Templates that are flexible in noting the type and extend of LLM usage are preferred. Templates that avoid unduly distracting or alarming the user are preferred. [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 19:56, 19 March 2026 (UTC)
::::Do we discuss here or there? I have replied you there as your proposal is about that policy so it is tradition to discuss it at the affected talk page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:59, 19 March 2026 (UTC)
: {{support}} Thanks for the proposed policy development and discussion; also note proposed policy talk page discussion: [[Wikiversity talk:Artificial intelligence]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:05, 24 March 2026 (UTC)
::I think the Wikiversity AI policy shall be official. – [[User:RestoreAccess111|RestoreAccess111]] &lt;sup style=&quot;font-family:Arimo, Arial;&quot;&gt;[[User talk:RestoreAccess111|Talk!]]&lt;/sup&gt; &lt;sup style=&quot;font-family:Times New Roman, Tinos;&quot;&gt;[[Special:Contributions/RestoreAccess111|Watch!]]&lt;/sup&gt; 06:11, 13 April 2026 (UTC)
{{archive bottom}}

== New titles for user right nominations ==
&lt;div class=&quot;cd-moveMark&quot;&gt;''Moved from [[Wikiversity talk:Candidates for Custodianship#New titles for user right nominations]].  [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:20, 17 April 2026 (UTC)''&lt;/div&gt;

I would like to propose the following retitles should a user be nominated for any of the following user rights:
* Curator: Candidates for Curatorship
* Bureaucrat: Candidates for Bureaucratship
The reason is that many curator (and probably bureaucrat) requests have run solely under {{tq|Candidates for Custodianship}}, but that title might sound misleading (especially in regards to the permission a user is requesting). CheckUser and Oversight (suppressor) are not included above since no user was nominated for these sensitive permissions, probably. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:30, 19 March 2026 (UTC)

:And it's not that when someone at the beginning misplaced the request, no one thought to move it and the others copied it. Even today, it would be possible to simply take it all and move it. Otherwise, for me, the more fundamental problem is that there is [[Wikiversity:Curators|no approved policy for curators]] than where the requests are based. Curators then operate in a certain vacuum and if one of them &quot;breaks out of the chain&quot;, the average user doesn't have many transparent tools to deal with it, because there is no policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:02, 19 March 2026 (UTC)
::I am not talking about the curator page (policy proposal). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:08, 21 March 2026 (UTC)
: @[[User:Juandev|Juandev]] I'll see if I can do an overhaul of [[Wikiversity:Candidates for Custodianship]], just like I recently did with the Requests for adminship page on English Wikiquote. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:17, 18 April 2026 (UTC)
:Yes, great idea - ideally there will be separate &quot;Candidates for ...&quot; pages for each user right group. The most important for now is to separate curator and custodian pages as CN suggests. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 1 May 2026 (UTC)
:So maybe I previously misunderstood. Are you proposing separated pages for nominations (i.e. [[Wikiversity:Candidates for Curatorship]], [[Wikiversity:Candidates for Bureaucratship]], [[Wikiversity:Candidates for Custodianship]])? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:30, 5 May 2026 (UTC)
:: Yes. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:33, 5 May 2026 (UTC)

== Technical Request:  Courtesy link.. ==

[[Template_talk:Information#Background_must_have_color_defined_as_well]] [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
: I can't edit the template directly as it need an sysop/interface admin to do it. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
:: Also if the Template field of - https://en.wikiversity.org/wiki/Special:LintErrors/night-mode-unaware-background-color is examined, there is poential for an admin to clear a substantial proportion of these by implmenting a simmilar fix to the indciated templates (and underlying stylesheets). It would be nice to clear things like Project box and others, as many other templates (and thus pages depend on them.) :)
[[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)

:I think it would be best to grant you interface admin rights for a short period of time to make these changes. However, I still have doubts about the suitability of this solution, which may cause other problems and no one has explained to me why dark mode has to be implemented this way @[[User:ShakespeareFan00|ShakespeareFan00]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:43, 20 March 2026 (UTC)

: I would have reservations about holding such rights, which is why I was trying to do what I could without needing them. However if it is the only way to get the required changes made, I would suggest asking on Wikipedia to find technical editors, willing to undertake the changes needed. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:32, 21 March 2026 (UTC)

== WikiEducator has closed ==

Some of you may know of a similar project to Wikiversity, called [https://wikieducator.org/Main_Page WikiEducator], championed by [https://oerfoundation.org/about/staff/wayne-mackintosh/ Wayne Mackintosh][https://www.linkedin.com/posts/waynemackintosh_important-notice-about-the-oer-foundation-activity-7405113051688931329-Nhm9/][https://openeducation.nz/killed-not-starved/].

It seems [https://openeducation.nz/terminating-oer-foundation their foundation has closed] and they are no longer operating.

They had done quite a bit of outreach (e.g., in the Pacific and Africa) to get educators using wiki.

The WikiEducator content is still available in MediaWiki - and potentially could be imported to Wikiversity ([https://wikieducator.org/WikiEducator:Copyrights CC-BY-SA] is the default license).

The closing of WikiEducator arguably makes the nurturing of Wikiversity even more important.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:09, 1 April 2026 (UTC)

:I was never active there. If anyone has an account or is otherwise in contact, we may want to copy relevant information here or even at [[:outreach:]]. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:46, 1 April 2026 (UTC)

:: I reached out to [[User:Mackiwg~enwikiversity|Wayne]] in January, and he responded briefly but positively (while travelling).  I wrote to the low-traffic wikieducator mailing list today and got a nice [https://groups.google.com/g/wikieducator/c/r_yIyUw6ZIA reply] from [[user:SteveFoerster|Steve Foerster]] who's interested in helping. If we can figure out a migration path it would be great to adopt at least the main namespace pages here.  
:: A few questions that come to mind: 
:: - would people want to create matching user accounts
:: - are there any namespaces (user/talk?) that should not be moved over
:: We could look at how this was done for the [[m:Wikivoyage/Migration]] wikivoyage migration. &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 04:27, 1 May 2026 (UTC)
:::That's fantastic, SJ, that you've reached out and that Wayne, Steve, and Jim are receptive—and that you can help! -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:52, 1 May 2026 (UTC)

== Wikinews is ending ==

Apparently mainly due to low editorial activity, low public interest, but also failure to achieve the goals from the proposal for the creation of the project, the Wikinews project is ending after years of discussions ([[Meta:Proposal for Closing Wikinews|some reading]]).

And I would be interested to see how Wikiversity is doing in the monitored metrics. We probably have more editors than Wikinews had, but what about consumers and achieving the goals? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:14, 1 April 2026 (UTC)

:Wikiversity's biggest issue in recent times was the hosting of low-quality, trash content. Thankfully we've done a great job in removing pseudoscience and other embarrassingly trash content (Wikidebates, for example), but the biggest concern moving forward is proper maintenance IMO. I've caught several pseudoscience pages being created within the last few months that could easily have flown under the radar (ex, [[The Kelemen Dilemma: Causal Collapse and Axiomatic Instability]]), so I'd urge our custodians/curators to be on the lookout for this type of content. Usually an AI-overview can point this type of content out relatively well.
:In terms of visibility, I believe Wikiversity is a high-traffic project. I remember my [[Mathematical Properties]] showing up on the first page of Google when searching up &quot;math properties&quot; for the longest time (and is still showing up in the first page 'till this day!). Besides, Wikinews hosted a lot of short-term content (the nature of news articles), while Wikiversity hosts content that can still be useful a decade later (ex, [[A Reader's Guide to Annotation]]). 
:I think we are on a better path than we were a few months ago, and I do want to thank everyone here who has been helping out with maintaining our website! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:48, 1 April 2026 (UTC)
:For what it's worth, the group that did that study has since disbanded, so no one is monitoring the other sister projects in the same way. Additionally, Wikinews had some catastrophic server issues due to the maintenance of [[:m:Extension:DynamicPageList]] which don't apply here. Your questions are still worth addressing, but I just wanted to cut off any concern at the pass about Wikiversity being in the same precarious situation. Wikiversity is definitely the biggest &quot;lagging behind&quot; or &quot;failure&quot; project now that Wikinews is being shuttered, but I don't see any near- or medium-term pathway to closing Wikiversity. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:46, 2 April 2026 (UTC)

:[[w:en:Wikipedia:Wikipedia Signpost/2026-03-31/News and notes|Entirety of Wikinews to be shut down]] (Wikipedia Signpost) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:03, 11 April 2026 (UTC)

== Action Required: Update templates/modules for electoral maps (Migrating from P1846 to P14226) ==

Hello everyone,

This is a notice regarding an ongoing data migration on Wikidata that may affect your election-related templates and Lua modules (such as &lt;code&gt;Module:Itemgroup/list&lt;/code&gt;).

'''The Change:'''&lt;br /&gt;
Currently, many templates pull electoral maps from Wikidata using the property [[:d:Property:P1846|P1846]], combined with the qualifier [[:d:Property:P180|P180]]: [[:d:Q19571328|Q19571328]]. 

We are migrating this data (across roughly 4,000 items) to a newly created, dedicated property: '''[[:d:Property:P14226|P14226]]'''.

'''What You Need To Do:'''&lt;br /&gt;
To ensure your templates and infoboxes do not break or lose their maps, please update your local code to fetch data from [[:d:Property:P14226|P14226]] instead of the old [[:d:Property:P1846|P1846]] + [[:d:Property:P180|P180]] structure. A [[m:Wikidata/Property Migration: P1846 to P14226/List|list of pages]] was generated using Wikimedia Global Search.

'''Deadline:'''&lt;br /&gt;
We are temporarily retaining the old data on [[:d:Property:P1846|P1846]] to allow for a smooth transition. However, to complete the data cleanup on Wikidata, the old [[:d:Property:P1846|P1846]] statements will be removed after '''May 1, 2026'''. Please update your modules and templates before this date to prevent any disruption to your wiki's election articles.

Let us know if you have any questions or need assistance with the query logic. Thank you for your help! [[User:ZI Jony|ZI Jony]] using [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 17:11, 3 April 2026 (UTC)
&lt;!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&amp;oldid=29941252 --&gt;
:I didnt find such properties, so we are probably fine. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:00, 12 April 2026 (UTC)
:: +1 (agreed). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:19, 12 April 2026 (UTC)

== Enable the abuse filter block action? ==

In light of [[Special:AbuseLog/80178]] (coupon spam), I would like to propose enabling the block action for the abuse filter. Only custodians will be able to enable and disable that action on an abuse filter, and it is useful to block ongoing vandalism. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:12, 13 April 2026 (UTC)

:Seems like a good idea, almost all of the users which create such pages are spambots so this shouldn’t be a problem. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:41, 13 April 2026 (UTC)
:Can you explain some more (I am new to abuse filters)? It looks like the attempted edit was prevented? Which abuse filter?
:Note on your suggestion, have also reactivated Antispam Filter 12 - see [[WV:RCA]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:45, 15 April 2026 (UTC)
:: I am proposing that we activate the abuse filter block action, which if a user triggers an abuse filter, it would actually block the user in question - the same mechanism that a custodian would use to block users. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:11, 15 April 2026 (UTC)
:::OK, thankyou, that makes sense. And, reviewing the abuse filter 12 log, it would be helpful because it would prevent the need for manual blocking. But I don't see a setting for autoblocking? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:14, 15 April 2026 (UTC)
:::: I think it probably adds an autoblock. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:43, 16 April 2026 (UTC)
: [[User:Jtneill|Jtneill]] and [[User:PieWriter|PieWriter]], given that a little bit more than a week has passed and there is minimal consensus to activate the abuse filter block action, I filed [[phab:T424053]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:05, 21 April 2026 (UTC)
::Thank-you for doing this. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:03, 24 April 2026 (UTC)

== Advice needed: A Neurodiversity-inspired Idea/observation ==

If I want the greatest participation of others to &quot;provide constructive criticism to my idea&quot; or to &quot;shoot down my idea&quot; or &quot;idea&quot;.

What I've called it so far is &quot;The Neurodiversity-inspired Idea&quot;. At other times I used more sensationalist wording but here on Wikiversity I don't dare do that. I actually woke up with thinking about putting this into my userspace draft: &quot;Personal Observations Made By Meeting Autistic and Non-Autistic Adults&quot;.

My ultimate goal is to stop blathering about my &quot;idea&quot; to friend and family without feeling my &quot;methodology&quot; is going into any progressive direction whatsoever. My latest encounter was somewhat constructive though. A friend of a friend who worked with people presenting ideas in attempting to getting grants. I don't want a grant. I just want to figure out how I can express my &quot;idea&quot; in a way so that I can more clearly figure out what flaws it got.

At the same time I tend to overthink. If anyone thinks etherpad might be a good place and considering Wikimedia already got an etherpad at https://etherpad.wikimedia.org/ if anyone feels like they know me better in the future feel free to suggest a &quot;session&quot; on etherpad.

'''If I don't receive a reply to this in 1 week's time I will begin to explore this &quot;idea&quot; into my userspace''' unless you replied and refrained me from doing so, of course. Then maybe after &quot;developing it there&quot; I might reference it to you another future time here in the Colloquium, with my &quot;idea&quot; still in my userspace draft. This &quot;idea&quot; is sort of a burden, I'm happy I've made the choice to get rid of it and hopefully move on with my life, unless there is something to this &quot;idea&quot;. 

My failure is probably evident: I feel I haven't told you anything. Same happened to when I talked to friends and family. In danger of overthinking it further I'll publish this right now. I need to &quot;keep it together&quot; [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:36, 16 April 2026 (UTC)

:Good on you putting it out there ... and hitting publish :). I'd say go for it (no need to wait), give birth to your idea and share about it here and elsewhere. Let it take shape and see where it might go. In many ways, this is exactly what an open collaborative learning community should be doing. Others might not know well how to respond, so perhaps consider creating some questions to accompany the idea. Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:21, 16 April 2026 (UTC)
::Thank you for encouraging me in developing the idea.
::I have created a &quot;questions&quot; section in the draft which is visible in the table of contents now. My brain was &quot;frozen&quot; today metaphorically speaking in that I felt I had like a &quot;writer's block&quot; so the draft has more &quot;AI/LLM&quot; content than before. I used the LLM for generating questions. The answers are so far human-only.
::I've also created a subsection where I could add the prompts that made the LLM generate the questions. That could help people make better prompts perhaps. I've described what it is about inside of it and there are some chaotically written notes.
::[[Draft:The_Neurodiversity-inspired_Idea#Questions_that_might_encourage_the_development_of_this_idea_and_its_methodology]]
::My draft is missing stuff. Any questions that you contribute to my draft will probably help me and if I don't understand the questions I'll probably notify you and also at the same time &quot;feed them&quot; to an LLM and ask in my input like &quot;explain in simple words what this question means, what is it searching for?&quot; etc. while I wait for an answer. If you have any more feedback please give it to me here or on the Draft page, its talk page or my user talk page. Thank you for helping me! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 21:20, 18 April 2026 (UTC)
::Today I woke up with not only thinking about supplying questions along with the &quot;idea&quot; but also answers. ie. Is it possible to &quot;test&quot; this idea? Is it possible to create one or multiple hypotheses based on this &quot;idea&quot;?(etc.) I've thought about this before in this &quot;idea&quot; but since I'm beginning to add to Wikiversity what was previously 'locked in my mind' it's also easier for me to see what I've done so far. Thank you for this comment! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 09:11, 23 April 2026 (UTC)
:May I think that you should not add deadlines ; being read, and rising interest for collaboration, or even simply for exchange of thoughts, such an effective meeting event loads a huge bunch of unprobability, which time can help to… somehow diminish. Maybe, I would advice you having a central place for developping your ideas, your needs, your advances, maybe a page in your own user zone, and from time to time, depending your feeling, it could be every trimester or so, or more frequently, you could write a short account of progress (or even of no progress), or a call for participation, in such a place as this present one ; I think that will increase much exposure of your projet. Maybe also, if you can find a project name, not necessarily very meaningfull by itseilf (at least it will gain signification with time, as your project develops), that will serve as a kind-of hook, and make your announcement titles more visible. Best regards (and my excuses for my poor command of English, which seems to be unplease an anti-abuse filter, &quot;Questionable Language (profanity)&quot;, which I don't understand…). My few cents. --  [[User:Eric.LEWIN|Eric.LEWIN]] ([[User talk:Eric.LEWIN|discussion]] • [[Special:Contributions/Eric.LEWIN|contributions]]) 10:06, 17 April 2026 (UTC)
::Sorry about the false positive on the profanity filter - I've fixed it. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:26, 17 April 2026 (UTC)
:::&quot;May I think that you should not add deadlines ; being read, and rising interest for collaboration, or even simply for exchange of thoughts, such an effective meeting event loads a huge bunch of unprobability, which time can help to… somehow diminish.&quot;
::Thank you Eric for this comment. Trust in time is how I interpret it. I should not feel like I need to be in a hurry. I'll try to give this time. Thank you!
:::&quot;Maybe, I would advice you having a central place for developping your ideas, your needs, your advances, maybe a page in your own user zone, and from time to time, depending your feeling, it could be every trimester or so, or more frequently, you could write a short account of progress (or even of no progress), or a call for participation, in such a place as this present one ; I think that will increase much exposure of your projet.&quot;
::A central place for developing or making &quot;project notes&quot; regarding the Neurodiversity idea on my userspace, I might need that, like a diary or &quot;project notes&quot; of the Neurodiversity idea similar to my course notes regarding my experience with Coursera.
::Any actions I take are going to be related to my Userspace from now on but I'll also update the draft when necessary. Now in the beginning I might be working daily to once every 3 days on both the draft and the daily notes I plan to make.
:::&quot;Maybe also, if you can find a project name, not necessarily very meaningfull by itseilf (at least it will gain signification with time, as your project develops), that will serve as a kind-of hook, and make your announcement titles more visible.&quot;
::Thank you for the advice. I was brainstorming yesterday about it. I concluded that since I've not yet developed a methodology that adheres to &quot;Do no harm&quot; and this is my first time working my &quot;idea&quot; into a way that is compatible with how projects develop on English Wikiversity this is new to me. My methodology isn't developed and therefore trying to get attention to my project through a name can wait. Yesterday I figured out a silly title that has nothing to do with the project: &quot;Planetary Awareness Potato Cabbage Rolls&quot; or something like that. Google output read that no such thing exists so I wanted it mainly to be unique. I don't want to raise attention that I'm unsure whether I'll actually be capable of developing a methodology for but project notes is my best bet so far in tracking my progress. Every day I think about this &quot;idea&quot; but I need to improve the important parts.
:::&quot;Best regards (and my excuses for my poor command of English, which seems to be unplease an anti-abuse filter, &quot;Questionable Language (profanity)&quot;, which I don't understand…). My few cents.&quot;
::You added great points and I felt that I was helped by you! I encourage you to post again and I can understand that interacting with any kind of automated filter can be discouraging and can be for me too! Thank you for giving me feedback! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 16:01, 18 April 2026 (UTC)

== Add some user rights to the curator user group? ==

By default, only custodians have the ability to mark new pages as patrolled (&lt;code&gt;patrol&lt;/code&gt;) and have their own page creations automatically marked as patrolled (&lt;code&gt;autopatrol&lt;/code&gt;). I am proposing both of the following:
* Curators can mark new pages as patrolled, helping on reducing the backlog of new, unpatrolled pages.
* New pages made by curators will be automatically marked as patrolled by the MediaWiki software.
Before we implement this, I would suggest implementing a proposed guideline for marking new pages as patrolled for curators and custodians.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:32, 17 April 2026 (UTC)

:Agree, &lt;s&gt;also can we also allow curators to undelete pages since they already have the rights to delete them?&lt;/s&gt; [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:54, 18 April 2026 (UTC)
::I think the requirement that undelete NOT be included came from above (meta / stewards / central office). Having access to the undelete page gives access to information that is restricted by their policies to admins (custodians and bureaucrats). -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:12, 18 April 2026 (UTC)
::: [[User:PieWriter|PieWriter]], unless if requests for curator and custodian should be RfA-like processes (that is, including voting and comments), then I have to agree with Dave above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:03, 18 April 2026 (UTC)
::::Oh, I didn’t realise that. Withdrawing my comment.. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:08, 19 April 2026 (UTC)
:{{support}} Seems reasonable and would reduce overhead. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 14:35, 18 April 2026 (UTC)
:'''Agree''', implement it also to [[Wikiversity:Curators]] proposal please. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:11, 18 April 2026 (UTC)
: I went ahead and filed [[phab:T424445]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:39, 26 April 2026 (UTC)

== [[Wikiversity:Curators|Curators and curators policy]] ==

How does it come, that Wikiversity has curators, but Curators policy is still being proposed? How do the curators exists and act if the policy about them havent been approved yet? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:33, 16 October 2025 (UTC)

:It looks as if it is not just curators. The policy on Bureaucratship is still being proposed as well. See [[Wikiversity:Bureaucratship]]. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 18:33, 27 October 2025 (UTC)

:I think its just the nature of a small WMF sister project in that there are lots of drafts, gaps, and potential improvements. In this case, these community would need to vote on those proposed Wikiversity staff policies if we think they're ready. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:08, 3 December 2025 (UTC)
:What? I thought you were getting it approved, Juandev... :) [[User:I&amp;#39;m Mr. Chris|I&amp;#39;m Mr. Chris]] ([[User talk:I&amp;#39;m Mr. Chris|discuss]] • [[Special:Contributions/I&amp;#39;m Mr. Chris|contribs]]) 14:20, 12 February 2026 (UTC)
::Yeah I think this one is important too and we need to aprove it too @[[User:I'm Mr. Chris|I'm Mr. Chris]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 12 February 2026 (UTC)
:::I thinks its ready to made into a policy, it seems to be complete and informative about what the rights does and how to get it. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:08, 15 February 2026 (UTC)
::::Agree -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:00, 27 March 2026 (UTC)

Let's make this the official discussion about adopting the [[Wikiversity:Curators|curators policy]] policy. Your comments are invited and welcome. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:40, 24 April 2026 (UTC)
: There were two similar Colloquium threads in separate places about the proposed curators policy. So I've moved them to be adjacent. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:42, 1 May 2026 (UTC)

== Wikiversity:Curators to become a policy ==

I've looked at the discussions about the Curators policy, I've looked at the practices, and it seems to me that there is no dispute about the wording of the policy, and what's more, the community has been using this proposal as if it were an offical policy for several years. Therefore, I propose that [[Wikiversity:Curators]] become a policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:35, 18 April 2026 (UTC)

:{{support}} —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 18 April 2026 (UTC)
:{{support}} —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:21, 18 April 2026 (UTC)
: {{support|Yes, please}}. Especially after when I and PieWriter proposed above, I agree. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:27, 24 April 2026 (UTC)
:: @[[User:Juandev|Juandev]]; as of now, curators now have the user rights &lt;code&gt;autopatrol&lt;/code&gt; and &lt;code&gt;patrol&lt;/code&gt;. Perhaps we should also include that in [[Wikiversity:Custodianship]]? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 12:07, 30 April 2026 (UTC)
:::You meant [[Wikiversity:Curators]] @[[User:Codename Noreste|Codename Noreste]]? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:15, 5 May 2026 (UTC)
:::: I agree that we must develop what rules curators should follow when marking new pages as patrolled; the same can be added for custodians since they can also mark new pages as patrolled. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:37, 5 May 2026 (UTC)
:{{support}} -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:42, 1 May 2026 (UTC)
:{{Support}} per nom. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 13:32, 1 May 2026 (UTC)

== Inactive curators ==

Hello, even though [[Wikiversity:Curators]] is not a policy yet, there are curators listed here that have been inactive for two years or more:
* {{user|Cody naccarato}} (last edit on 13 Dec 2022, last logged action on 10 Dec 2022)
* {{user|Praxidicae}} (last edit on 10 Sep 2022, last logged action on 12 Sep 2022)
[[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 19 April 2026 (UTC)

:Yup, I would remove the rights. To get the rights back if theyll come back should not be a big deal. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:08, 24 April 2026 (UTC)
:: When they don't reply by May 19, feel free (or any custodian) to do so. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:28, 25 April 2026 (UTC)

== Is anyone interested in Neurodiversity? ==

Is anyone interested in Neurodiversity? Is there anyone here who is interested for Neurodiversity to be &quot;something more&quot; than it already is? Does anyone here consider Neurodiversity one of the &quot;harder topics&quot; to work on or discuss? Does anyone here have an opinion about the [[Neurodiversity Movement]]? So these questions don't appear like &quot;out of a vacuum&quot; I can tell you a bit about my background:

Many years ago I got a psychiatric diagnosis &quot;Asperger's&quot;. After I stepped out of the office and my Äsperger's was 'concluded', I stepped out into the street and thought my first negative thought(but the positive thought followed after). The thought was about concentration camps in the second world war and that the world seemed to be going into the direction of &quot;labeling others&quot;. I was unsure whether this was &quot;real science&quot; and sort of &quot;challenged myself&quot; to make up my own mind after meeting people that had been given this diagnosis. The more adults with this diagnosis I met the more I started seeing &quot;patterns&quot;.

Was it a coincidence that the first person with Asperger's I met reminded me about my father later after I had plenty of times of experience with interacting with him? None of the people I interacted with online through IRC text chat...I felt I got any clue about how &quot;their brains work&quot;. Only when I met one person from the Asperger's chat community in person we both realized that whatever we experienced was akin to the &quot;chaos theory&quot;. He told me about &quot;chaos theory&quot; while I didn't know even what that term meant but I guess I 'read between the lines'. My question that I linger on still today is &quot;did he understand about me what I think I understood about him?&quot;? That our brains had the same configuration? Most autistic adults who meet other autistic adults usually get disappointed. They think the diagnosis will help them meet somebody like themselves and then they realize the great diversity in the autistic spectrum created by Psychiatry.

I later stopped interacting with autistic communities that much, I felt that it did not benefit me. Also Neurodiversity's &quot;neurotypes&quot; interested me for a while until I realized I had &quot;misunderstood everything&quot; about them and how they are used in the Neurodiversity Movement or &quot;Neurodiversity community&quot; if that even can precisely be defined? I doubt it but if you want to contribute to the [[Neurodiversity Movement]]. My previous attempts failed as I got more and more confused. I think a community project needs a community. With a lack of that I don't think it is worth my time. If any of you would like to work on that project let me know on my talk page.

So I was kinda lost and was talking to my friend and psychologist and I realized if I never talk about my idea to anyone in a &quot;comprehensive way&quot; or show that it matters to me nothing is going to ever happen. So I started talking about my &quot;idea&quot; more. Nobody could understand the &quot;idea&quot; because I had not developed my skills regarding where to start...although the process had already started &quot;automatically&quot; and that's why I often think of &quot;well my brain sort of activated me&quot;. I don't feel like I did have a plan and this idea happened. It happened &quot;by itself&quot;. My brain reacted to what I was seeing in a video or stream.

I value interaction highly in this idea. I think it would be helpful to make a community of people who are not paranoid about stuff that can express itself like &quot;don't analyze me!&quot;, &quot;don't compare me to anyone!&quot;. 

On the contrary, more often than not those adults who were diagnosed were actually openly comparing themselves with each other and I think that is healthy in a &quot;science&quot; way if done the &quot;right way&quot; which probably means &quot;Do no harm&quot;.

I found video material is important but I'm very unsure if uploading own video material to Wikimedia Commons would constitute a &quot;reasonable&quot; use of the resources there. Maybe somebody here needs to ask more questions to me that I should answer before that happens. I also know the '''be bold''' so I could just do what I think might be ok. Though I work better in a group as long as I know what &quot;group configurations&quot; help me. This is in a non-profit way. Since the state supported me this might be a way I am trying to &quot;give back&quot; to the state and &quot;the world&quot;. May seem overly ambitious and crazy but this thing gives me energy. It gives me hope when trying to develop this idea. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:47, 23 April 2026 (UTC)

:Thanks for sharing. There is plenty of room for neurodiversity community learning. However, the challenge I think is that the intersection of those interested in (a) ND, and (b) English Wikiversity might be very small (e.g., 1!) at this point in time.
:But don't give up hope. For example, Wikipedia has many more ND-interested editors; maybe consider reaching out to see who might be interested:
:[[w:Category:Wikipedians interested in neurodiversity]]
:You could also start an equivalent category here:
:[[:Category:Wikiversitarians interested in neurodiversity]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:46, 6 May 2026 (UTC)

== Request for comment (global AI policy) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;A [[:m:Requests for comment/Artificial intelligence policy|request for comment]] is currently being held to decide on a global AI policy. {{int:Feedback-thanks-title}} [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 00:58, 26 April 2026 (UTC)&lt;/bdi&gt;
&lt;!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30424282 --&gt;

== Coming over From wikinews ==

Any chance someone could help me if you are allowed to write news articles here since wikinews is going read only mode soon, thank you! [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:43, 1 May 2026 (UTC)

:The scope of Wikiversity is very broad and is basically about more-or-less any learning material. We have made it a point to not have duplicative content of other WMF projects, but since Wikinews is being shuttered, I personally am fine with writing news articles here. One thing that is not controversial at all is a learning resource &lt;em&gt;about&lt;/em&gt; how to write news: that could be hugely useful here and could involve the process of writing news stories to learn and to share back and forth with an editor or fact-checker. In fact, I'd support an entire namespace dedicated to keeping the notion of Wikinews alive here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:38, 1 May 2026 (UTC)
::Thank you so much! How do I start? Cheers! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:07, 2 May 2026 (UTC)
:::I think it's premature to start just making news articles en masse, but if you want to start discussing the topic of citizen journalism, you can do that now. [[:Category:Journalism]] already has some material, so you can start by seeing what we already have, how you can refine that, etc. You can definitely have learning resources with collaborators who want to learn about journalism ASAP. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:24, 2 May 2026 (UTC)
::::thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:38, 2 May 2026 (UTC)
::::If I could try and start one News Article could you please tell me how to go about it? Like what style of writing like Wikinews or something else? Thank you Justin! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:48, 2 May 2026 (UTC)
:::::Honestly, there are very few policies and guidelines here. I think the best way to write a news story would be in a manner that is obvious and instructive. So, for instance, it's common to use the &quot;pyramid style&quot; when you're writing news, so if you were to write a story that makes it very clear that you are using that approach, that would be helpful. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:08, 2 May 2026 (UTC)
::::::cool thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 02:13, 2 May 2026 (UTC)

== Language learning  ==

toki! I am trying to add or see what the toki pona language learning stuff on here is but I don't see anything that is language learning for anything. [[User:Jan Imon|Jan Imon]] ([[User talk:Jan Imon|discuss]] • [[Special:Contributions/Jan Imon|contribs]]) 23:13, 2 May 2026 (UTC) —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:29, 3 May 2026 (UTC)

:We have language materials ([[:Category:Languages]], [[World Languages]], [[Portal:Foreign Language Learning]], [[Portal:Multilingual Studies]]). They are not as developed as I think we would all like and there's not any coverage of Toki Pona, but in principle, we could and would like that. You can also see [[:b:Subject:Languages]] at our sister project Wikibooks. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:33, 3 May 2026 (UTC)

== Timeline format? ==

I’ve been working on the World War II articles, including the [[World War II/Timeline|timeline]], and is there a specific timeline format that should be used? Right now it’s just a table, and there’s no separation between different periods/phases of the war.

I don’t want to use [[mw:Extension:EasyTimeline]] because this will be displaying dates and not time periods. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 01:35, 4 May 2026 (UTC)

:I dont think we have a policy or guideline, how to format a timeline. But you may try to browes wikiversity by Google if someone was dealing with this in the past somewhow @[[User:PhilDaBirdMan|PhilDaBirdMan]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:23, 5 May 2026 (UTC)
::+1 - there's no specific guideline on how to format a timeline, it's really up to you. In my opinion I think the timeline is good. I'd personally bold the dates just to make it easier to separate it from the event description, but that's my personal 2 cents. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:18, 5 May 2026 (UTC)
:::I’ll probably remove links to the dates/years, they’re just Wikipedia pages that shouldn’t be over linked to. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 00:39, 6 May 2026 (UTC)

== Interface administrator for Codename Noreste ==

Hello, everyone. I am requesting interface administrator access on this wiki.

The main reasoning is that I would benefit from having the user right &lt;code&gt;editinterface&lt;/code&gt;, which would allow me to make dark mode changes to pages in the MediaWiki namespace, add &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-parser-output&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt; to some interface pages using templates, handle interface-protected edit requests, and similar stuff. Additionally, I have some knowledge of CSS, and I would like to assist with modifying CSS pages whenever necessary, such as moving MediaWiki common.css code to TemplateStyles CSS pages.

I am requesting the maximum time that is allowed per the [[Wikiversity:Interface administrators|policy]], and I have 2FA enabled on my account. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:55, 6 May 2026 (UTC)

*{{support}} Globally trusted user. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:07, 6 May 2026 (UTC)
*{{support}} Trusted and knowledgeable. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:35, 6 May 2026 (UTC)
*{{support}} WV would benefit from this. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 08:32, 6 May 2026 (UTC)
*{{support}} --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:13, 7 May 2026 (UTC)</text>
      <sha1>8d3tjmarm6y8zng6chilv38dbziln5l</sha1>
    </revision>
    <revision>
      <id>2807864</id>
      <parentid>2807862</parentid>
      <timestamp>2026-05-07T09:20:20Z</timestamp>
      <contributor>
        <username>Juandev</username>
        <id>2651</id>
      </contributor>
      <comment>/* Wikiversity:Curators to become a policy */ Reply</comment>
      <origin>2807864</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="76500" sha1="d05s4n26xba76if6k4d0slzhyeipqp4" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;
== Requested update to [[Wikiversity:Interface administrators]] ==

Currently, [[Wikiversity:Interface administrators]] is a policy that includes a caveat that interface admins are not required long-term and that user right can only be added for a period of up to two weeks. I am proposing that we remove this qualification and allow for indefinite interface admin status. I think this is useful because there are reasons for tweaking the site CSS or JavaScript (e.g. to comply with dark mode), add gadgets (e.g. importing Cat-a-Lot, which I would like to do), or otherwise modifying the site that could plausibly come up on an irregular basis and requiring the overhead of a bureaucrat to add the user rights is inefficient. In particular, I am also going to request this right if the community accepts indefinite interface admins. Thoughts? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:23, 17 August 2025 (UTC)

:And who will then monitor them to make sure they don't damage the project in any way, or abuse the rights acquired in this way? For large projects, this might not be a problem, but for smaller projects like the English Wikiversity, I'm not sure if there are enough users who would say, something is happening here that shouldn't be happening. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:28, 20 August 2025 (UTC)
::Anyone would be who. This argument applies to any person with any advanced rights here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 10:46, 20 August 2025 (UTC)
:I think it is reasonable to allow for longer periods of access than 2 weeks to interface admin and support adjusting the policy to allow for this flexibility. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:57, 2 December 2025 (UTC)
::+1 —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:38, 25 January 2026 (UTC)
:@[[User:Koavf|Koavf]] I agree that the two-week requirement could be revised, but wouldn’t people just request access for a specific purpose anyway? Instead of granting indefinite access, they should request the specific time frame they need the rights for—until the planned fixes are completed—and then request an extension if more time is required. We could remove the two-week criterion while still keeping the access explicitly temporary. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:48, 25 January 2026 (UTC)
::I just don't see why this wiki needs to be different than all of the others. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:18, 25 January 2026 (UTC)
:::There isn’t really much of a need for a permanent one at this point in time [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 09:53, 25 January 2026 (UTC)
:I quite agree with this proposal, so long as they perform the suggested changes as mentioned here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 04:06, 26 January 2026 (UTC)
:: Just to clarify, I support '''indefinite interface admin status'''. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:34, 13 April 2026 (UTC)
:I think there is decent consensus for lengthening this, but not necessarily for indefinite permissions, so does anyone object to me revising it to the standard being 120 days instead of two weeks? I'll check back on this thread in three weeks and if there's no objection, I'll make the change. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:47, 13 April 2026 (UTC)
::Sure [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:27, 13 April 2026 (UTC)
::Thanks for proposing this, Justin. I agree with the proposal to lengthen the interface admin period from 2 weeks but not indefinitely. Can I check the source(s) for the standard being 120 days (I'm guessing policies on other projects or maybe global policy?)? In any case, I think it is reasonable for us to adopt a similar period. However, note on the current policy discussion page notes from @[[User:Dave Braunschweig|Dave Braunschweig]] arguing for shorter periods to lower risk, that's why it is 2 weeks. But if there are projects that need longer access, that should also be accommodated. Maybe we could adjust the policy to specify that ''interface admin rights can be given for 14 to 120 days depending on how long is required and what is supported by the community''. Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:29, 24 April 2026 (UTC)
:::There was there was no source for 120: it was just more than 14 and less than infinity. The &quot;14 to 120&quot; also seems reasonable. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 14:33, 24 April 2026 (UTC)
::: On some small/medium-sized wikis, such as English Wikibooks and English Wikiquote for example, indefinite interface administrator access for administrators is allowed, but they tend not to make changes to the CSS and JS page changes unless it's truly necessary. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:34, 24 April 2026 (UTC)
:::It's a good idea to make the length of this right on request or allow to be prolonged. However, IA should test large changes somewhere else, for example on the en.wv mirror, and only after testing it on the mirror, adapt it to the live version. That means I can't imagine a time-consuming operation right now. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:04, 24 April 2026 (UTC)
::::Sorry, what mirror is this? Are you talking about beta.wv? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 20:32, 24 April 2026 (UTC)
:::::Not beta.wv. Basically somewhere else then on a live wiki. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:59, 24 April 2026 (UTC)
:::::: Wouldn't testing on a user's own common.css page work anyway? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:36, 24 April 2026 (UTC)
:Change made here: https://en.wikiversity.org/w/index.php?title=Wikiversity%3AInterface_administrators&amp;diff=2807543&amp;oldid=2806289 —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 13:35, 4 May 2026 (UTC)

== [[Template:AI-generated]] ==

After going through the plethora of ChatGPT-generated pages made by [[User:Lbeaumont|Lbeaumont]] (with many more pages to go), I'd like community input on this proposal to [[Wikiversity:Artificial intelligence]] that I think would be benefical for the community:
*Resources generated by AI '''must''' be indicated as so through the project box, [[Template:AI-generated]], on either the page or the main resource (if the page is a part of a project).
I do not believe including a small note/reference that a page is AI-generated is sufficient, and I take my thinking from [[WV:Original research|Wikiversity's OR policy]] for OR work: ''Within Wikiversity, all original research should be clearly identified as such''. I believe resources created from AI should also be clearly indicated as such, especially since we are working on whether or not AI-generated resources should be allowed on the website (discussion is [[Wikiversity talk:Artificial intelligence|here]], for reference). This makes it easier for organizational purposes, and in the event ''if'' we ban AI-generated work. 

I've left a message on Lee's talk page over a week ago and did not get a response or acknowledgement, so I'd like for the community's input for this inclusion to the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:53, 26 January 2026 (UTC)

:I believe that existing Wikiversity policies are sufficient. Authors are responsible for the accuracy and usefulness of the content that is published. This policy covers AI-generated content that is: 1) carefully reviewed by the author publishing it, and 2) the source is noted.   [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 19:38, 27 January 2026 (UTC)
::A small reference for pages that are substantially filled with Chat-GPT entries, like [[Real Good Religion]], [[Attributing Blame]], [[Fostering Curiosity]], are not sufficient IMO and a project box would be the best indicator that a page is AI-generated (especially when there is a mixture of human created content AND AI-generated content, as present in a lot of your pages). This is useful, especially considering the notable issues with AI (including hallucinations and fabrication of details), so viewers and support staff are aware. These small notes left on the pages are not as easily viewable as a project box or banner would be. I really don't see the issue with a clear-label guideline. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:34, 27 January 2026 (UTC)
::{{ping|Lbeaumont}} I noticed your reversions [https://en.wikiversity.org/w/index.php?title=Exploring_Existential_Concerns&amp;diff=prev&amp;oldid=2788278 here] &amp; [https://en.wikiversity.org/w/index.php?title=Subjective_Awareness&amp;diff=prev&amp;oldid=2788257 here]. I'd prefer to have a clean conversation regarding this proposition. Please voice your concerns here. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:53, 28 January 2026 (UTC)
:::Regarding Subjective Awareness, I distinctly recall the effort I went to to write that the old-fashioned way. It is true that ChatGPT assisted me in augmenting the list of words suggested as candidate subjective states. This is a small section of the course, is clearly marked, and makes no factual claim. Marking the entire course as AI-generated is misleading. I would have made these comments when I reverted your edit; however, the revert button does not provide that opportunity.
:::Regarding the Exploring Existential Concerns course, please note this was adapted from my EmotionalCompetency.com website, which predates the availability of LLMs. The course does include two links, clearly labeled as ChatGPT-generated. Again, marking the entire course as AI-generated is misleading.
:::On a broader issue, I don't consider your opinions to have established a carefully debated and adopted Wikiversity policy. You went ahead and modified many of my courses over my clearly stated objections. Please let this issue play out more completely before editing my courses further. Thanks.   [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 15:11, 29 January 2026 (UTC)
::::Understood, and I respect your position. I apologize if my edits were seen as overarching. We could change the project box to &quot;a portion of this resource was generated by AI&quot;, or something along those lines. Feel free to revert my changes where you see fit, and I encourage more users to provide their input. EDIT: I've made changes to the template to indicate that a portion of the content has been generated from an LLM. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:50, 29 January 2026 (UTC)
:::::Thanks for this reply. The new banner is unduly large and alarming. There is no need for alarm here. The use of AI is not harmful per se. Like any technology, it can be used to help or to harm. I take care to craft prompts carefully, point the LMM to reliable source materials, and to carefully read and verify the generated text before I publish it. This is all in keeping with long-established Wikiversity policy. We don't want to use a  [[w:One-drop_rule|one-drop rule]] here or cause a [[w:Satanic_panic|satanic panic]]. We can learn our lessons from history here. I don't see any pedagogical reason for establishing a classification of &quot;AI generated&quot;, but if there is a consensus that it is needed, perhaps it can be handled as just another category that learning resources can be assigned to. I would rather focus on identifying any errors in factual claims than on casting pejorative bias toward AI-generated content. An essay on the best practices for using LMM on Wikiveristy would be welcome.   [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 15:58, 30 January 2026 (UTC)
::::::The new banner mimics the banner that is available on the English Wikibooks (see [[b:Template:AI-generated]] &amp; [[b:Template:Uses AI]]), so my revisions aren't unique in this aspect. At this point, I'd welcome other peoples' inputs. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:40, 30 January 2026 (UTC)

== How do I start making pages? ==
Is there a notability guideline for Wikiversity? What is the sourcing policy for information? What is the Manual of Style? What kind of educational content qualifies for Wikiversity? All the introduction pages are a bit unclear. 
[[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 02:25, 28 January 2026 (UTC)

:{{ping|VidanaliK}} Welcome to Wikiversity! I've left you a welcome message on your talk page. That should help you out. Make sure to especially look at [[Wikiversity:Introduction]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 03:11, 28 January 2026 (UTC)
::It says that I can't post more pages because I have apparently exceeded the new page limit. How long does it take before that new page limit expires? [[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 16:57, 28 January 2026 (UTC)
:::This is a restriction for new users so that Wikiversity is not hit with massive spam. As for when this limit will expire, it should be a few days or after a certain number of edits. It's easy to overcome, though I do not have the exact numbers atm. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:08, 29 January 2026 (UTC)
::::OK, I think I got past the limit. [[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 17:21, 29 January 2026 (UTC)
==Why does it feel like Wikiversity is no longer really active anymore?==
I've been looking at recent changes, and both today and yesterday there haven't been many changes that I haven't made; it feels like walking through a ghost town, is this just me or is Wikiversity not really active anymore? [[User:VidanaliK|VidanaliK]] ([[User talk:VidanaliK|discuss]] • [[Special:Contributions/VidanaliK|contribs]]) 03:54, 30 January 2026 (UTC)

:There is fewer people editing these days compared to the past. Many newcomers tend to edit in Wikipedia instead. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 06:39, 30 January 2026 (UTC)
:It’s a little slow, but I’m happy to know that Wikiversity is a place that I think should provide value even if the activity of editors fluctuates. If it’s any consolation your edits may be encouraging for some anonymous newcomer to start edits on their own! I think it’s hard to build community when there is such a wide variety of interests and a smaller starting userbase. Also sometimes the getting into a particular topic that already exists can be intimidating because some relics (large portals, school, categories, etc.) have intricate, unique and generally messy levels of organization. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 22:16, 9 March 2026 (UTC)
:I'd say it comes down to working hard for Wikiversity, basically if somebody or a group of people will start presenting good ideas and they turn out to be provably stable.
:I even asked Google's &quot;AI Mode&quot;, what is Wikiversity famous for? Unfortunately it could not answer that.
:Simply, we have not made Wikiversity famous by presenting really provable stable ideas yet. My hope is that this time might come. Perhaps even this year 2026!
:Hope dies last. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:12, 27 April 2026 (UTC)

== Inactivity policy for Curators ==

I was wondering if there is a specific inactivity polity for curators (semi-admins) as I am pretty sure the global policy does not apply to them as they are not ''fully'' sysops. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:20, 15 February 2026 (UTC)

:Unfortunately, I don't see an inactivity policy, but if we were to create such a new policy for curators, it should be the same for custodians (administrators). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 18:45, 15 February 2026 (UTC)
::@[[User:Codename Noreste|Codename Noreste]] There is currently none, that I could find, for custodians either. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:47, 17 February 2026 (UTC)
:::I think we should propose a local inactivity policy for custodians (and by extension, curators), which should be at least one year without any edits ''and'' logged actions. However, I don't know which page should it be when the inactivity removal procedure starts. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:53, 17 February 2026 (UTC)
::::@[[User:Codename Noreste|Codename Noreste]] In theory, there should be a section added at [[WV:Candidates for custodianship]] [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:55, 17 February 2026 (UTC)
::::: To be consistent with the [[meta:Admin activity review|global period of 2 years inactivity]] for en.wv  [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship#How are bureaucrats removed?|Bureaucrats]] we could add something like this to [[Wikiversity:Curators]]:
::::::The maximum time period of inactivity &lt;u&gt;without community review&lt;/u&gt; for curators is two years (consistent with  the [[:meta:Category:Global policies|global policy]] described at [[meta:Admin activity review|Admin activity review]] which applies for [[Wikiversity:Custodianship#Notes|Custodians]] and [[Wikiversity:Bureaucratship|Bureaucrats]]). After that time a custodian will remove the rights.
::::: -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:51, 27 March 2026 (UTC)
:::::Yup, I agree with Jtneill, there is a policy proposal for Wikiversity:Curators, where it should be logically deployed. The question is if we are ready to aprove the policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:43, 17 April 2026 (UTC)
:::::: I agree, but we should notify the colloquium about inactive curators, just like a steward would do for inactive custodians and bureaucrats per [[:m:Admin activity review|AAR]]. What is the minimum timeframe an inactive curator should receive so they can respond they would keep their rights? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:49, 17 April 2026 (UTC)
:I incorporated these suggestions into the proposed curators policy. Please review/comment/improve. Summary: 2 years, notify curator's user page, then remove rights after 1 month: [[Wikiversity:Curators#Inactivity]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:59, 24 April 2026 (UTC)
:: @[[User:Jtneill|Jtneill]] I created [[Template:Inactive curator]] for this. Feel free to make any changes or improvements. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:29, 24 April 2026 (UTC)

== [[Wikiversity:Artificial intelligence]] to become an official policy ==
{{Archive top|After running for a week, there is consensus, alongside comments, for [[Wikiversity:Artificial intelligence]] to be implemented as an official policy. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:27, 17 April 2026 (UTC)}}
With the introduction of AI-material, and some material just plain disruptive, its imperative that Wikiversity catches up with its sister projects and implements an official AI policy that we can work with. The recent issue of [[User:Lbeaumont|Lbeaumont]]'s 50+ articles that contain significantly large AI-generated material has made me came to the Colloquium. This user has also been removing the [[Template:AI-generated]] template from their pages, calling it &quot;misleading&quot;, &quot;alarmist&quot;, and &quot;pejorative&quot; - which is all just simply nonsensical rationales. Not to even mention this user's contributions to the English Wikipedia have been [https://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Inner_Development_Goals contested] and [https://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Multipolar_trap removed] a couple of times (for being low-quality and clearly LLM-generated), highlighting the need for an actual policy to be implemented here on Wikiversity. I would like to ping {{ping|Juandev}} and {{ping|Jtneill}} for their thoughts as well, since I'd like this to be implemented as soon as possible. 

Wikiversity has a significant issue with implementing anti-disruptive measures, hence why we have received numerous complaints as a community about our quality. I originally was reverting the removal of the templates, but realized that this is still a proposed policy, which it shouldn't be anymore. It should be a recognized Wikiversity policy. 14:54, 10 March 2026 (UTC) —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:54, 10 March 2026 (UTC)

:@[[User:Atcovi|Atcovi]] '''I agree''' that the draft, should become official policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:00, 10 March 2026 (UTC)
:I provided a detailed response at: [[Wikiversity talk:Artificial intelligence#Evolving a Wikiversity policy on AI]]
:I will appreaciate it if you consder that carefully. [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 22:49, 10 March 2026 (UTC)
:Agree it should become official Wikiversity policy on the condition &lt;u&gt;that point point 5 is about [significant/substantial] LLM-generated text specifically&lt;/u&gt;. Not a good idea to overuse it, it should be added when there is substantial AI-generated text on the page, not for other cases. [[User:Prototyperspective|Prototyperspective]] ([[User talk:Prototyperspective|discuss]] • [[Special:Contributions/Prototyperspective|contribs]]) 12:37, 11 March 2026 (UTC)
:What policy is being debated? Is it the text on this page, which is pointed to by the general banner, or the text at:   [[Wikiversity:Artificial intelligence|Wikiversity:Artificial intelligence,]]   which is pointed to by the specific banner? Let's begin with coherence on the text being debated. Thanks!  [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 11:49, 17 March 2026 (UTC)
::@[[User:Lbeaumont|Lbeaumont]] This is a call for approval of the new Wikiversity policy. You expressed your opinion [[Wikiversity talk:Artificial intelligence#Evolving a Wikiversity policy on AI|on the talk page of the proposal]], I replied to you and await your response.When creating policies, it is necessary to propose specific solutions. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:12, 17 March 2026 (UTC)
:::Toward a Justified and Parsimonious AI Policy 
:::As we collaborate to develop a consensus policy on the use of Large Language Models, it is wise to begin by considering the needs of the various stakeholders to the policy.
:::The stakeholders are:
:::1)     The users, 
:::2)     The source providers, and
:::3)     The editors
:::There may also be others with a minor stake in this policy, including the population at large.
:::The many needs of the users are currently addressed by long-standing [[Wikiversity:Policies|Wikiversity policies]], so we can focus on what, if any, additional needs arise as LLMs are deployed. 
:::As always, users need assurance that propositional statements are accurate. This is covered by the existing policy on [[Wikiversity:Verifiability|verifiably]]. In addition, it is expected by both the users and those that provide materials used as sources for the text are [[Wikiversity:Cite sources|accurately attributed]]. This is also covered by [[Wikiversity:Cite sources|existing policies]].
:::To respect the time and effort of editors, a parsimonious policy will unburden editors from costly requirements that exceed benefits to the users.
:::Finally, it is important to recognize that because attention is our most valuable seizing attention unnecessarily is a form of theft. 
:::The following proposed policy statement results from these considerations:
:::Recommended Policy statement:
:::·       Editors [[Wikiversity:Verifiability|verify the accuracy]] of propositional statements, regardless of the source.
:::·       Editors [[Wikiversity:Cite sources|attribute the source]] of propositional statements. In the case of LLM, cite the LLM model and the prompt used.
:::·       Use of various available templates to mark the use of LLM are optional. Templates that are flexible in noting the type and extend of LLM usage are preferred. Templates that avoid unduly distracting or alarming the user are preferred. [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 19:56, 19 March 2026 (UTC)
::::Do we discuss here or there? I have replied you there as your proposal is about that policy so it is tradition to discuss it at the affected talk page. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:59, 19 March 2026 (UTC)
: {{support}} Thanks for the proposed policy development and discussion; also note proposed policy talk page discussion: [[Wikiversity talk:Artificial intelligence]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:05, 24 March 2026 (UTC)
::I think the Wikiversity AI policy shall be official. – [[User:RestoreAccess111|RestoreAccess111]] &lt;sup style=&quot;font-family:Arimo, Arial;&quot;&gt;[[User talk:RestoreAccess111|Talk!]]&lt;/sup&gt; &lt;sup style=&quot;font-family:Times New Roman, Tinos;&quot;&gt;[[Special:Contributions/RestoreAccess111|Watch!]]&lt;/sup&gt; 06:11, 13 April 2026 (UTC)
{{archive bottom}}

== New titles for user right nominations ==
&lt;div class=&quot;cd-moveMark&quot;&gt;''Moved from [[Wikiversity talk:Candidates for Custodianship#New titles for user right nominations]].  [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:20, 17 April 2026 (UTC)''&lt;/div&gt;

I would like to propose the following retitles should a user be nominated for any of the following user rights:
* Curator: Candidates for Curatorship
* Bureaucrat: Candidates for Bureaucratship
The reason is that many curator (and probably bureaucrat) requests have run solely under {{tq|Candidates for Custodianship}}, but that title might sound misleading (especially in regards to the permission a user is requesting). CheckUser and Oversight (suppressor) are not included above since no user was nominated for these sensitive permissions, probably. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:30, 19 March 2026 (UTC)

:And it's not that when someone at the beginning misplaced the request, no one thought to move it and the others copied it. Even today, it would be possible to simply take it all and move it. Otherwise, for me, the more fundamental problem is that there is [[Wikiversity:Curators|no approved policy for curators]] than where the requests are based. Curators then operate in a certain vacuum and if one of them &quot;breaks out of the chain&quot;, the average user doesn't have many transparent tools to deal with it, because there is no policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:02, 19 March 2026 (UTC)
::I am not talking about the curator page (policy proposal). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:08, 21 March 2026 (UTC)
: @[[User:Juandev|Juandev]] I'll see if I can do an overhaul of [[Wikiversity:Candidates for Custodianship]], just like I recently did with the Requests for adminship page on English Wikiquote. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:17, 18 April 2026 (UTC)
:Yes, great idea - ideally there will be separate &quot;Candidates for ...&quot; pages for each user right group. The most important for now is to separate curator and custodian pages as CN suggests. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:39, 1 May 2026 (UTC)
:So maybe I previously misunderstood. Are you proposing separated pages for nominations (i.e. [[Wikiversity:Candidates for Curatorship]], [[Wikiversity:Candidates for Bureaucratship]], [[Wikiversity:Candidates for Custodianship]])? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:30, 5 May 2026 (UTC)
:: Yes. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:33, 5 May 2026 (UTC)

== Technical Request:  Courtesy link.. ==

[[Template_talk:Information#Background_must_have_color_defined_as_well]] [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
: I can't edit the template directly as it need an sysop/interface admin to do it. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)
:: Also if the Template field of - https://en.wikiversity.org/wiki/Special:LintErrors/night-mode-unaware-background-color is examined, there is poential for an admin to clear a substantial proportion of these by implmenting a simmilar fix to the indciated templates (and underlying stylesheets). It would be nice to clear things like Project box and others, as many other templates (and thus pages depend on them.) :)
[[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 11:43, 20 March 2026 (UTC)

:I think it would be best to grant you interface admin rights for a short period of time to make these changes. However, I still have doubts about the suitability of this solution, which may cause other problems and no one has explained to me why dark mode has to be implemented this way @[[User:ShakespeareFan00|ShakespeareFan00]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:43, 20 March 2026 (UTC)

: I would have reservations about holding such rights, which is why I was trying to do what I could without needing them. However if it is the only way to get the required changes made, I would suggest asking on Wikipedia to find technical editors, willing to undertake the changes needed. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 09:32, 21 March 2026 (UTC)

== WikiEducator has closed ==

Some of you may know of a similar project to Wikiversity, called [https://wikieducator.org/Main_Page WikiEducator], championed by [https://oerfoundation.org/about/staff/wayne-mackintosh/ Wayne Mackintosh][https://www.linkedin.com/posts/waynemackintosh_important-notice-about-the-oer-foundation-activity-7405113051688931329-Nhm9/][https://openeducation.nz/killed-not-starved/].

It seems [https://openeducation.nz/terminating-oer-foundation their foundation has closed] and they are no longer operating.

They had done quite a bit of outreach (e.g., in the Pacific and Africa) to get educators using wiki.

The WikiEducator content is still available in MediaWiki - and potentially could be imported to Wikiversity ([https://wikieducator.org/WikiEducator:Copyrights CC-BY-SA] is the default license).

The closing of WikiEducator arguably makes the nurturing of Wikiversity even more important.

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:09, 1 April 2026 (UTC)

:I was never active there. If anyone has an account or is otherwise in contact, we may want to copy relevant information here or even at [[:outreach:]]. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:46, 1 April 2026 (UTC)

:: I reached out to [[User:Mackiwg~enwikiversity|Wayne]] in January, and he responded briefly but positively (while travelling).  I wrote to the low-traffic wikieducator mailing list today and got a nice [https://groups.google.com/g/wikieducator/c/r_yIyUw6ZIA reply] from [[user:SteveFoerster|Steve Foerster]] who's interested in helping. If we can figure out a migration path it would be great to adopt at least the main namespace pages here.  
:: A few questions that come to mind: 
:: - would people want to create matching user accounts
:: - are there any namespaces (user/talk?) that should not be moved over
:: We could look at how this was done for the [[m:Wikivoyage/Migration]] wikivoyage migration. &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 04:27, 1 May 2026 (UTC)
:::That's fantastic, SJ, that you've reached out and that Wayne, Steve, and Jim are receptive—and that you can help! -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:52, 1 May 2026 (UTC)

== Wikinews is ending ==

Apparently mainly due to low editorial activity, low public interest, but also failure to achieve the goals from the proposal for the creation of the project, the Wikinews project is ending after years of discussions ([[Meta:Proposal for Closing Wikinews|some reading]]).

And I would be interested to see how Wikiversity is doing in the monitored metrics. We probably have more editors than Wikinews had, but what about consumers and achieving the goals? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 19:14, 1 April 2026 (UTC)

:Wikiversity's biggest issue in recent times was the hosting of low-quality, trash content. Thankfully we've done a great job in removing pseudoscience and other embarrassingly trash content (Wikidebates, for example), but the biggest concern moving forward is proper maintenance IMO. I've caught several pseudoscience pages being created within the last few months that could easily have flown under the radar (ex, [[The Kelemen Dilemma: Causal Collapse and Axiomatic Instability]]), so I'd urge our custodians/curators to be on the lookout for this type of content. Usually an AI-overview can point this type of content out relatively well.
:In terms of visibility, I believe Wikiversity is a high-traffic project. I remember my [[Mathematical Properties]] showing up on the first page of Google when searching up &quot;math properties&quot; for the longest time (and is still showing up in the first page 'till this day!). Besides, Wikinews hosted a lot of short-term content (the nature of news articles), while Wikiversity hosts content that can still be useful a decade later (ex, [[A Reader's Guide to Annotation]]). 
:I think we are on a better path than we were a few months ago, and I do want to thank everyone here who has been helping out with maintaining our website! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:48, 1 April 2026 (UTC)
:For what it's worth, the group that did that study has since disbanded, so no one is monitoring the other sister projects in the same way. Additionally, Wikinews had some catastrophic server issues due to the maintenance of [[:m:Extension:DynamicPageList]] which don't apply here. Your questions are still worth addressing, but I just wanted to cut off any concern at the pass about Wikiversity being in the same precarious situation. Wikiversity is definitely the biggest &quot;lagging behind&quot; or &quot;failure&quot; project now that Wikinews is being shuttered, but I don't see any near- or medium-term pathway to closing Wikiversity. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 00:46, 2 April 2026 (UTC)

:[[w:en:Wikipedia:Wikipedia Signpost/2026-03-31/News and notes|Entirety of Wikinews to be shut down]] (Wikipedia Signpost) -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:03, 11 April 2026 (UTC)

== Action Required: Update templates/modules for electoral maps (Migrating from P1846 to P14226) ==

Hello everyone,

This is a notice regarding an ongoing data migration on Wikidata that may affect your election-related templates and Lua modules (such as &lt;code&gt;Module:Itemgroup/list&lt;/code&gt;).

'''The Change:'''&lt;br /&gt;
Currently, many templates pull electoral maps from Wikidata using the property [[:d:Property:P1846|P1846]], combined with the qualifier [[:d:Property:P180|P180]]: [[:d:Q19571328|Q19571328]]. 

We are migrating this data (across roughly 4,000 items) to a newly created, dedicated property: '''[[:d:Property:P14226|P14226]]'''.

'''What You Need To Do:'''&lt;br /&gt;
To ensure your templates and infoboxes do not break or lose their maps, please update your local code to fetch data from [[:d:Property:P14226|P14226]] instead of the old [[:d:Property:P1846|P1846]] + [[:d:Property:P180|P180]] structure. A [[m:Wikidata/Property Migration: P1846 to P14226/List|list of pages]] was generated using Wikimedia Global Search.

'''Deadline:'''&lt;br /&gt;
We are temporarily retaining the old data on [[:d:Property:P1846|P1846]] to allow for a smooth transition. However, to complete the data cleanup on Wikidata, the old [[:d:Property:P1846|P1846]] statements will be removed after '''May 1, 2026'''. Please update your modules and templates before this date to prevent any disruption to your wiki's election articles.

Let us know if you have any questions or need assistance with the query logic. Thank you for your help! [[User:ZI Jony|ZI Jony]] using [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 17:11, 3 April 2026 (UTC)
&lt;!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&amp;oldid=29941252 --&gt;
:I didnt find such properties, so we are probably fine. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:00, 12 April 2026 (UTC)
:: +1 (agreed). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:19, 12 April 2026 (UTC)

== Enable the abuse filter block action? ==

In light of [[Special:AbuseLog/80178]] (coupon spam), I would like to propose enabling the block action for the abuse filter. Only custodians will be able to enable and disable that action on an abuse filter, and it is useful to block ongoing vandalism. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:12, 13 April 2026 (UTC)

:Seems like a good idea, almost all of the users which create such pages are spambots so this shouldn’t be a problem. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 23:41, 13 April 2026 (UTC)
:Can you explain some more (I am new to abuse filters)? It looks like the attempted edit was prevented? Which abuse filter?
:Note on your suggestion, have also reactivated Antispam Filter 12 - see [[WV:RCA]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:45, 15 April 2026 (UTC)
:: I am proposing that we activate the abuse filter block action, which if a user triggers an abuse filter, it would actually block the user in question - the same mechanism that a custodian would use to block users. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:11, 15 April 2026 (UTC)
:::OK, thankyou, that makes sense. And, reviewing the abuse filter 12 log, it would be helpful because it would prevent the need for manual blocking. But I don't see a setting for autoblocking? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:14, 15 April 2026 (UTC)
:::: I think it probably adds an autoblock. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:43, 16 April 2026 (UTC)
: [[User:Jtneill|Jtneill]] and [[User:PieWriter|PieWriter]], given that a little bit more than a week has passed and there is minimal consensus to activate the abuse filter block action, I filed [[phab:T424053]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:05, 21 April 2026 (UTC)
::Thank-you for doing this. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:03, 24 April 2026 (UTC)

== Advice needed: A Neurodiversity-inspired Idea/observation ==

If I want the greatest participation of others to &quot;provide constructive criticism to my idea&quot; or to &quot;shoot down my idea&quot; or &quot;idea&quot;.

What I've called it so far is &quot;The Neurodiversity-inspired Idea&quot;. At other times I used more sensationalist wording but here on Wikiversity I don't dare do that. I actually woke up with thinking about putting this into my userspace draft: &quot;Personal Observations Made By Meeting Autistic and Non-Autistic Adults&quot;.

My ultimate goal is to stop blathering about my &quot;idea&quot; to friend and family without feeling my &quot;methodology&quot; is going into any progressive direction whatsoever. My latest encounter was somewhat constructive though. A friend of a friend who worked with people presenting ideas in attempting to getting grants. I don't want a grant. I just want to figure out how I can express my &quot;idea&quot; in a way so that I can more clearly figure out what flaws it got.

At the same time I tend to overthink. If anyone thinks etherpad might be a good place and considering Wikimedia already got an etherpad at https://etherpad.wikimedia.org/ if anyone feels like they know me better in the future feel free to suggest a &quot;session&quot; on etherpad.

'''If I don't receive a reply to this in 1 week's time I will begin to explore this &quot;idea&quot; into my userspace''' unless you replied and refrained me from doing so, of course. Then maybe after &quot;developing it there&quot; I might reference it to you another future time here in the Colloquium, with my &quot;idea&quot; still in my userspace draft. This &quot;idea&quot; is sort of a burden, I'm happy I've made the choice to get rid of it and hopefully move on with my life, unless there is something to this &quot;idea&quot;. 

My failure is probably evident: I feel I haven't told you anything. Same happened to when I talked to friends and family. In danger of overthinking it further I'll publish this right now. I need to &quot;keep it together&quot; [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:36, 16 April 2026 (UTC)

:Good on you putting it out there ... and hitting publish :). I'd say go for it (no need to wait), give birth to your idea and share about it here and elsewhere. Let it take shape and see where it might go. In many ways, this is exactly what an open collaborative learning community should be doing. Others might not know well how to respond, so perhaps consider creating some questions to accompany the idea. Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:21, 16 April 2026 (UTC)
::Thank you for encouraging me in developing the idea.
::I have created a &quot;questions&quot; section in the draft which is visible in the table of contents now. My brain was &quot;frozen&quot; today metaphorically speaking in that I felt I had like a &quot;writer's block&quot; so the draft has more &quot;AI/LLM&quot; content than before. I used the LLM for generating questions. The answers are so far human-only.
::I've also created a subsection where I could add the prompts that made the LLM generate the questions. That could help people make better prompts perhaps. I've described what it is about inside of it and there are some chaotically written notes.
::[[Draft:The_Neurodiversity-inspired_Idea#Questions_that_might_encourage_the_development_of_this_idea_and_its_methodology]]
::My draft is missing stuff. Any questions that you contribute to my draft will probably help me and if I don't understand the questions I'll probably notify you and also at the same time &quot;feed them&quot; to an LLM and ask in my input like &quot;explain in simple words what this question means, what is it searching for?&quot; etc. while I wait for an answer. If you have any more feedback please give it to me here or on the Draft page, its talk page or my user talk page. Thank you for helping me! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 21:20, 18 April 2026 (UTC)
::Today I woke up with not only thinking about supplying questions along with the &quot;idea&quot; but also answers. ie. Is it possible to &quot;test&quot; this idea? Is it possible to create one or multiple hypotheses based on this &quot;idea&quot;?(etc.) I've thought about this before in this &quot;idea&quot; but since I'm beginning to add to Wikiversity what was previously 'locked in my mind' it's also easier for me to see what I've done so far. Thank you for this comment! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 09:11, 23 April 2026 (UTC)
:May I think that you should not add deadlines ; being read, and rising interest for collaboration, or even simply for exchange of thoughts, such an effective meeting event loads a huge bunch of unprobability, which time can help to… somehow diminish. Maybe, I would advice you having a central place for developping your ideas, your needs, your advances, maybe a page in your own user zone, and from time to time, depending your feeling, it could be every trimester or so, or more frequently, you could write a short account of progress (or even of no progress), or a call for participation, in such a place as this present one ; I think that will increase much exposure of your projet. Maybe also, if you can find a project name, not necessarily very meaningfull by itseilf (at least it will gain signification with time, as your project develops), that will serve as a kind-of hook, and make your announcement titles more visible. Best regards (and my excuses for my poor command of English, which seems to be unplease an anti-abuse filter, &quot;Questionable Language (profanity)&quot;, which I don't understand…). My few cents. --  [[User:Eric.LEWIN|Eric.LEWIN]] ([[User talk:Eric.LEWIN|discussion]] • [[Special:Contributions/Eric.LEWIN|contributions]]) 10:06, 17 April 2026 (UTC)
::Sorry about the false positive on the profanity filter - I've fixed it. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:26, 17 April 2026 (UTC)
:::&quot;May I think that you should not add deadlines ; being read, and rising interest for collaboration, or even simply for exchange of thoughts, such an effective meeting event loads a huge bunch of unprobability, which time can help to… somehow diminish.&quot;
::Thank you Eric for this comment. Trust in time is how I interpret it. I should not feel like I need to be in a hurry. I'll try to give this time. Thank you!
:::&quot;Maybe, I would advice you having a central place for developping your ideas, your needs, your advances, maybe a page in your own user zone, and from time to time, depending your feeling, it could be every trimester or so, or more frequently, you could write a short account of progress (or even of no progress), or a call for participation, in such a place as this present one ; I think that will increase much exposure of your projet.&quot;
::A central place for developing or making &quot;project notes&quot; regarding the Neurodiversity idea on my userspace, I might need that, like a diary or &quot;project notes&quot; of the Neurodiversity idea similar to my course notes regarding my experience with Coursera.
::Any actions I take are going to be related to my Userspace from now on but I'll also update the draft when necessary. Now in the beginning I might be working daily to once every 3 days on both the draft and the daily notes I plan to make.
:::&quot;Maybe also, if you can find a project name, not necessarily very meaningfull by itseilf (at least it will gain signification with time, as your project develops), that will serve as a kind-of hook, and make your announcement titles more visible.&quot;
::Thank you for the advice. I was brainstorming yesterday about it. I concluded that since I've not yet developed a methodology that adheres to &quot;Do no harm&quot; and this is my first time working my &quot;idea&quot; into a way that is compatible with how projects develop on English Wikiversity this is new to me. My methodology isn't developed and therefore trying to get attention to my project through a name can wait. Yesterday I figured out a silly title that has nothing to do with the project: &quot;Planetary Awareness Potato Cabbage Rolls&quot; or something like that. Google output read that no such thing exists so I wanted it mainly to be unique. I don't want to raise attention that I'm unsure whether I'll actually be capable of developing a methodology for but project notes is my best bet so far in tracking my progress. Every day I think about this &quot;idea&quot; but I need to improve the important parts.
:::&quot;Best regards (and my excuses for my poor command of English, which seems to be unplease an anti-abuse filter, &quot;Questionable Language (profanity)&quot;, which I don't understand…). My few cents.&quot;
::You added great points and I felt that I was helped by you! I encourage you to post again and I can understand that interacting with any kind of automated filter can be discouraging and can be for me too! Thank you for giving me feedback! [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 16:01, 18 April 2026 (UTC)

== Add some user rights to the curator user group? ==

By default, only custodians have the ability to mark new pages as patrolled (&lt;code&gt;patrol&lt;/code&gt;) and have their own page creations automatically marked as patrolled (&lt;code&gt;autopatrol&lt;/code&gt;). I am proposing both of the following:
* Curators can mark new pages as patrolled, helping on reducing the backlog of new, unpatrolled pages.
* New pages made by curators will be automatically marked as patrolled by the MediaWiki software.
Before we implement this, I would suggest implementing a proposed guideline for marking new pages as patrolled for curators and custodians.

Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:32, 17 April 2026 (UTC)

:Agree, &lt;s&gt;also can we also allow curators to undelete pages since they already have the rights to delete them?&lt;/s&gt; [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 02:54, 18 April 2026 (UTC)
::I think the requirement that undelete NOT be included came from above (meta / stewards / central office). Having access to the undelete page gives access to information that is restricted by their policies to admins (custodians and bureaucrats). -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 20:12, 18 April 2026 (UTC)
::: [[User:PieWriter|PieWriter]], unless if requests for curator and custodian should be RfA-like processes (that is, including voting and comments), then I have to agree with Dave above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 22:03, 18 April 2026 (UTC)
::::Oh, I didn’t realise that. Withdrawing my comment.. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 00:08, 19 April 2026 (UTC)
:{{support}} Seems reasonable and would reduce overhead. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 14:35, 18 April 2026 (UTC)
:'''Agree''', implement it also to [[Wikiversity:Curators]] proposal please. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:11, 18 April 2026 (UTC)
: I went ahead and filed [[phab:T424445]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:39, 26 April 2026 (UTC)

== [[Wikiversity:Curators|Curators and curators policy]] ==

How does it come, that Wikiversity has curators, but Curators policy is still being proposed? How do the curators exists and act if the policy about them havent been approved yet? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:33, 16 October 2025 (UTC)

:It looks as if it is not just curators. The policy on Bureaucratship is still being proposed as well. See [[Wikiversity:Bureaucratship]]. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 18:33, 27 October 2025 (UTC)

:I think its just the nature of a small WMF sister project in that there are lots of drafts, gaps, and potential improvements. In this case, these community would need to vote on those proposed Wikiversity staff policies if we think they're ready. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:08, 3 December 2025 (UTC)
:What? I thought you were getting it approved, Juandev... :) [[User:I&amp;#39;m Mr. Chris|I&amp;#39;m Mr. Chris]] ([[User talk:I&amp;#39;m Mr. Chris|discuss]] • [[Special:Contributions/I&amp;#39;m Mr. Chris|contribs]]) 14:20, 12 February 2026 (UTC)
::Yeah I think this one is important too and we need to aprove it too @[[User:I'm Mr. Chris|I'm Mr. Chris]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 12 February 2026 (UTC)
:::I thinks its ready to made into a policy, it seems to be complete and informative about what the rights does and how to get it. [[User:PieWriter|PieWriter]] ([[User talk:PieWriter|discuss]] • [[Special:Contributions/PieWriter|contribs]]) 03:08, 15 February 2026 (UTC)
::::Agree -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:00, 27 March 2026 (UTC)

Let's make this the official discussion about adopting the [[Wikiversity:Curators|curators policy]] policy. Your comments are invited and welcome. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 08:40, 24 April 2026 (UTC)
: There were two similar Colloquium threads in separate places about the proposed curators policy. So I've moved them to be adjacent. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:42, 1 May 2026 (UTC)

== Wikiversity:Curators to become a policy ==

I've looked at the discussions about the Curators policy, I've looked at the practices, and it seems to me that there is no dispute about the wording of the policy, and what's more, the community has been using this proposal as if it were an offical policy for several years. Therefore, I propose that [[Wikiversity:Curators]] become a policy. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:35, 18 April 2026 (UTC)

:{{support}} —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:54, 18 April 2026 (UTC)
:{{support}} —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 20:21, 18 April 2026 (UTC)
: {{support|Yes, please}}. Especially after when I and PieWriter proposed above, I agree. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 14:27, 24 April 2026 (UTC)
:: @[[User:Juandev|Juandev]]; as of now, curators now have the user rights &lt;code&gt;autopatrol&lt;/code&gt; and &lt;code&gt;patrol&lt;/code&gt;. Perhaps we should also include that in [[Wikiversity:Custodianship]]? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 12:07, 30 April 2026 (UTC)
:::You meant [[Wikiversity:Curators]] @[[User:Codename Noreste|Codename Noreste]]? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:15, 5 May 2026 (UTC)
:::: I agree that we must develop what rules curators should follow when marking new pages as patrolled; the same can be added for custodians since they can also mark new pages as patrolled. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 13:37, 5 May 2026 (UTC)
:::::I see, well I think you can just add this to the policy. It is not major change and it probably reflects actual practice or actual technical possibilities for those flags. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:20, 7 May 2026 (UTC)
:{{support}} -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 12:42, 1 May 2026 (UTC)
:{{Support}} per nom. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 13:32, 1 May 2026 (UTC)

== Inactive curators ==

Hello, even though [[Wikiversity:Curators]] is not a policy yet, there are curators listed here that have been inactive for two years or more:
* {{user|Cody naccarato}} (last edit on 13 Dec 2022, last logged action on 10 Dec 2022)
* {{user|Praxidicae}} (last edit on 10 Sep 2022, last logged action on 12 Sep 2022)
[[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 19 April 2026 (UTC)

:Yup, I would remove the rights. To get the rights back if theyll come back should not be a big deal. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:08, 24 April 2026 (UTC)
:: When they don't reply by May 19, feel free (or any custodian) to do so. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:28, 25 April 2026 (UTC)

== Is anyone interested in Neurodiversity? ==

Is anyone interested in Neurodiversity? Is there anyone here who is interested for Neurodiversity to be &quot;something more&quot; than it already is? Does anyone here consider Neurodiversity one of the &quot;harder topics&quot; to work on or discuss? Does anyone here have an opinion about the [[Neurodiversity Movement]]? So these questions don't appear like &quot;out of a vacuum&quot; I can tell you a bit about my background:

Many years ago I got a psychiatric diagnosis &quot;Asperger's&quot;. After I stepped out of the office and my Äsperger's was 'concluded', I stepped out into the street and thought my first negative thought(but the positive thought followed after). The thought was about concentration camps in the second world war and that the world seemed to be going into the direction of &quot;labeling others&quot;. I was unsure whether this was &quot;real science&quot; and sort of &quot;challenged myself&quot; to make up my own mind after meeting people that had been given this diagnosis. The more adults with this diagnosis I met the more I started seeing &quot;patterns&quot;.

Was it a coincidence that the first person with Asperger's I met reminded me about my father later after I had plenty of times of experience with interacting with him? None of the people I interacted with online through IRC text chat...I felt I got any clue about how &quot;their brains work&quot;. Only when I met one person from the Asperger's chat community in person we both realized that whatever we experienced was akin to the &quot;chaos theory&quot;. He told me about &quot;chaos theory&quot; while I didn't know even what that term meant but I guess I 'read between the lines'. My question that I linger on still today is &quot;did he understand about me what I think I understood about him?&quot;? That our brains had the same configuration? Most autistic adults who meet other autistic adults usually get disappointed. They think the diagnosis will help them meet somebody like themselves and then they realize the great diversity in the autistic spectrum created by Psychiatry.

I later stopped interacting with autistic communities that much, I felt that it did not benefit me. Also Neurodiversity's &quot;neurotypes&quot; interested me for a while until I realized I had &quot;misunderstood everything&quot; about them and how they are used in the Neurodiversity Movement or &quot;Neurodiversity community&quot; if that even can precisely be defined? I doubt it but if you want to contribute to the [[Neurodiversity Movement]]. My previous attempts failed as I got more and more confused. I think a community project needs a community. With a lack of that I don't think it is worth my time. If any of you would like to work on that project let me know on my talk page.

So I was kinda lost and was talking to my friend and psychologist and I realized if I never talk about my idea to anyone in a &quot;comprehensive way&quot; or show that it matters to me nothing is going to ever happen. So I started talking about my &quot;idea&quot; more. Nobody could understand the &quot;idea&quot; because I had not developed my skills regarding where to start...although the process had already started &quot;automatically&quot; and that's why I often think of &quot;well my brain sort of activated me&quot;. I don't feel like I did have a plan and this idea happened. It happened &quot;by itself&quot;. My brain reacted to what I was seeing in a video or stream.

I value interaction highly in this idea. I think it would be helpful to make a community of people who are not paranoid about stuff that can express itself like &quot;don't analyze me!&quot;, &quot;don't compare me to anyone!&quot;. 

On the contrary, more often than not those adults who were diagnosed were actually openly comparing themselves with each other and I think that is healthy in a &quot;science&quot; way if done the &quot;right way&quot; which probably means &quot;Do no harm&quot;.

I found video material is important but I'm very unsure if uploading own video material to Wikimedia Commons would constitute a &quot;reasonable&quot; use of the resources there. Maybe somebody here needs to ask more questions to me that I should answer before that happens. I also know the '''be bold''' so I could just do what I think might be ok. Though I work better in a group as long as I know what &quot;group configurations&quot; help me. This is in a non-profit way. Since the state supported me this might be a way I am trying to &quot;give back&quot; to the state and &quot;the world&quot;. May seem overly ambitious and crazy but this thing gives me energy. It gives me hope when trying to develop this idea. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:47, 23 April 2026 (UTC)

:Thanks for sharing. There is plenty of room for neurodiversity community learning. However, the challenge I think is that the intersection of those interested in (a) ND, and (b) English Wikiversity might be very small (e.g., 1!) at this point in time.
:But don't give up hope. For example, Wikipedia has many more ND-interested editors; maybe consider reaching out to see who might be interested:
:[[w:Category:Wikipedians interested in neurodiversity]]
:You could also start an equivalent category here:
:[[:Category:Wikiversitarians interested in neurodiversity]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:46, 6 May 2026 (UTC)

== Request for comment (global AI policy) ==

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;A [[:m:Requests for comment/Artificial intelligence policy|request for comment]] is currently being held to decide on a global AI policy. {{int:Feedback-thanks-title}} [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 00:58, 26 April 2026 (UTC)&lt;/bdi&gt;
&lt;!-- Message sent by User:Codename Noreste@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=30424282 --&gt;

== Coming over From wikinews ==

Any chance someone could help me if you are allowed to write news articles here since wikinews is going read only mode soon, thank you! [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 22:43, 1 May 2026 (UTC)

:The scope of Wikiversity is very broad and is basically about more-or-less any learning material. We have made it a point to not have duplicative content of other WMF projects, but since Wikinews is being shuttered, I personally am fine with writing news articles here. One thing that is not controversial at all is a learning resource &lt;em&gt;about&lt;/em&gt; how to write news: that could be hugely useful here and could involve the process of writing news stories to learn and to share back and forth with an editor or fact-checker. In fact, I'd support an entire namespace dedicated to keeping the notion of Wikinews alive here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:38, 1 May 2026 (UTC)
::Thank you so much! How do I start? Cheers! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:07, 2 May 2026 (UTC)
:::I think it's premature to start just making news articles en masse, but if you want to start discussing the topic of citizen journalism, you can do that now. [[:Category:Journalism]] already has some material, so you can start by seeing what we already have, how you can refine that, etc. You can definitely have learning resources with collaborators who want to learn about journalism ASAP. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:24, 2 May 2026 (UTC)
::::thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:38, 2 May 2026 (UTC)
::::If I could try and start one News Article could you please tell me how to go about it? Like what style of writing like Wikinews or something else? Thank you Justin! @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 01:48, 2 May 2026 (UTC)
:::::Honestly, there are very few policies and guidelines here. I think the best way to write a news story would be in a manner that is obvious and instructive. So, for instance, it's common to use the &quot;pyramid style&quot; when you're writing news, so if you were to write a story that makes it very clear that you are using that approach, that would be helpful. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:08, 2 May 2026 (UTC)
::::::cool thanks. [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 02:13, 2 May 2026 (UTC)

== Language learning  ==

toki! I am trying to add or see what the toki pona language learning stuff on here is but I don't see anything that is language learning for anything. [[User:Jan Imon|Jan Imon]] ([[User talk:Jan Imon|discuss]] • [[Special:Contributions/Jan Imon|contribs]]) 23:13, 2 May 2026 (UTC) —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:29, 3 May 2026 (UTC)

:We have language materials ([[:Category:Languages]], [[World Languages]], [[Portal:Foreign Language Learning]], [[Portal:Multilingual Studies]]). They are not as developed as I think we would all like and there's not any coverage of Toki Pona, but in principle, we could and would like that. You can also see [[:b:Subject:Languages]] at our sister project Wikibooks. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:33, 3 May 2026 (UTC)

== Timeline format? ==

I’ve been working on the World War II articles, including the [[World War II/Timeline|timeline]], and is there a specific timeline format that should be used? Right now it’s just a table, and there’s no separation between different periods/phases of the war.

I don’t want to use [[mw:Extension:EasyTimeline]] because this will be displaying dates and not time periods. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 01:35, 4 May 2026 (UTC)

:I dont think we have a policy or guideline, how to format a timeline. But you may try to browes wikiversity by Google if someone was dealing with this in the past somewhow @[[User:PhilDaBirdMan|PhilDaBirdMan]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:23, 5 May 2026 (UTC)
::+1 - there's no specific guideline on how to format a timeline, it's really up to you. In my opinion I think the timeline is good. I'd personally bold the dates just to make it easier to separate it from the event description, but that's my personal 2 cents. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 14:18, 5 May 2026 (UTC)
:::I’ll probably remove links to the dates/years, they’re just Wikipedia pages that shouldn’t be over linked to. [[User:PhilDaBirdMan|PhilDaBirdMan]] ([[User talk:PhilDaBirdMan|discuss]] • [[Special:Contributions/PhilDaBirdMan|contribs]]) 00:39, 6 May 2026 (UTC)

== Interface administrator for Codename Noreste ==

Hello, everyone. I am requesting interface administrator access on this wiki.

The main reasoning is that I would benefit from having the user right &lt;code&gt;editinterface&lt;/code&gt;, which would allow me to make dark mode changes to pages in the MediaWiki namespace, add &lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-parser-output&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt; to some interface pages using templates, handle interface-protected edit requests, and similar stuff. Additionally, I have some knowledge of CSS, and I would like to assist with modifying CSS pages whenever necessary, such as moving MediaWiki common.css code to TemplateStyles CSS pages.

I am requesting the maximum time that is allowed per the [[Wikiversity:Interface administrators|policy]], and I have 2FA enabled on my account. Thank you. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 00:55, 6 May 2026 (UTC)

*{{support}} Globally trusted user. ―[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''&lt;span style=&quot;color:black&quot;&gt;v&lt;/span&gt;f&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 01:07, 6 May 2026 (UTC)
*{{support}} Trusted and knowledgeable. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 04:35, 6 May 2026 (UTC)
*{{support}} WV would benefit from this. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 08:32, 6 May 2026 (UTC)
*{{support}} --[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:13, 7 May 2026 (UTC)</text>
      <sha1>d05s4n26xba76if6k4d0slzhyeipqp4</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Sandbox</title>
    <ns>4</ns>
    <id>1558</id>
    <revision>
      <id>2807830</id>
      <parentid>2806317</parentid>
      <timestamp>2026-05-06T17:39:47Z</timestamp>
      <contributor>
        <username>~2026-27532-56</username>
        <id>3070757</id>
      </contributor>
      <origin>2807830</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2169" sha1="rbu0zuxauvuekc6juckrvtu9i3widdi" xml:space="preserve">{{/header template}}
== PROJECT: BUSINESS RESURRECTION PLAN (TEMPLATE) ==

'''Inihanda ni:''' [[User:ILAGAY_ANG_USERNAME_DITO|ANG PANGALAN MO]]
'''Kumpanyang Pinili:''' [Pangalan ng Kumpanya - Hal. Solid Video, Nokia, o Kodak]
'''Petsa:''' {{CURRENTMONTHNAME}} {{CURRENTDAY}}, {{CURRENTYEAR}}

---

== 1. CASE STUDY AUDIT (Bakit kami nalugi?) ==
Batay sa aking research sa [[Wikipedia]], narito ang mga dahilan kung bakit nahirapan ang kumpanyang ito sa kanilang Information System:

* '''Legacy Issue:''' [Hal. Masyadong umaasa sa analog film/physical rentals.]
* '''Failure to Adapt:''' [Hal. Hindi agad nag-shift sa streaming o digital cloud storage.]
* '''Competitor Edge:''' [Hal. Ang Netflix/Android ay mas mabilis at mas mura ang system.]

---

== 2. PROPOSED MAINTENANCE &amp; ADAPTATION PLAN ==
Kung ako ang magiging Operations Manager noon, ito ang aking gagawing steps para hindi malugi ang kumpanya:

=== A. System Upgrade (Evolutionary Maintenance) ===
Ito ang mga bagong teknolohiya na dapat nating i-adopt:
# '''Digital Transition:''' [Hal. Magtatayo ng sariling website at mobile app.]
# '''Cloud Integration:''' [Hal. I-store ang lahat ng data sa cloud para hindi mawala.]

=== B. Preventive Maintenance (Monthly Checklist) ===
Upang masiguradong laging online ang negosyo, narito ang aming routine:
* '''Data Backup:''' Isasagawa ang backup tuwing [Oras/Araw] para iwas-loss ng customer records.
* '''Security Patching:''' Update ng firewall buwan-buwan para iwas-hack.
* '''User Feedback Audit:''' Pakikinig sa reklamo ng users tungkol sa bilis ng system.

---

== 3. BUSINESS CONTINUITY (The &quot;Plan B&quot;) ==
Kapag nag-crash ang main system, ito ang aming gagawin:
* '''Backup Server:''' Gagamit ng redundant servers para kung down ang isa, up ang kabila.
* '''Emergency Protocol:''' [Hal. Magpapadala ng email sa lahat ng customers sa loob ng 10 minutes.]

---

== 4. CONCLUSION ==
Ang tamang '''Information System Operation and Maintenance''' ay hindi lang gastos; ito ay investment. Kung nag-evolve ang aming system sa tamang panahon, hindi sana kami mapag-iiwanan ng kompetisyon. zefira sundar hai।.

[[Category:Business Information Systems Projects]]</text>
      <sha1>rbu0zuxauvuekc6juckrvtu9i3widdi</sha1>
    </revision>
    <revision>
      <id>2807856</id>
      <parentid>2807830</parentid>
      <timestamp>2026-05-07T05:46:36Z</timestamp>
      <contributor>
        <username>HannahTadea</username>
        <id>3070835</id>
      </contributor>
      <comment>Undo all revisions. Resource is empty, but not [[Wikiversity:Deletions|deleted]].</comment>
      <origin>2807856</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="0" sha1="phoiac9h4m842xq45sp7s6u21eteeq1" />
      <sha1>phoiac9h4m842xq45sp7s6u21eteeq1</sha1>
    </revision>
    <revision>
      <id>2807857</id>
      <parentid>2807856</parentid>
      <timestamp>2026-05-07T05:48:19Z</timestamp>
      <contributor>
        <username>HannahTadea</username>
        <id>3070835</id>
      </contributor>
      <origin>2807857</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2534" sha1="cyz5ym4byvh6sq2y9sxshl8dzpkiptv" xml:space="preserve">== The Unbreakable Business Plan ==

=== Business Name ===
SwiftCart Online Shop

=== Introduction ===
This Business Maintenance Plan is designed to protect the company’s operations, customer information, and online services from technical problems such as system crashes, hacking, and data loss. The goal of this plan is to ensure that the business can continue operating smoothly even during emergencies.

=== Standard Operating Procedure (SOP) ===

==== 1. Backup Schedule ====
* Customer and transaction data must be backed up every hour.
* A full system backup must be performed every day at 12:00 midnight.
* Backup files must be stored in cloud storage and an external hard drive.

'''Purpose:'''
This prevents the loss of important customer and business information during system failures.

==== 2. Security Check ====
* All employees must use strong passwords with letters, numbers, and symbols.
* Two-factor authentication must be enabled for all admin accounts.
* The system must be checked weekly for viruses and suspicious activities.
* Customer information must remain confidential and protected.

'''Purpose:'''
This helps prevent hacking, data theft, and unauthorized access.

==== 3. System Maintenance ====
* The website and application must be checked every morning before operations begin.
* Software updates must be installed immediately after testing.
* Broken links, payment errors, and loading problems must be fixed immediately.

'''Purpose:'''
Regular maintenance keeps the system fast, stable, and reliable.

==== 4. Emergency Response Plan ====
# Inform the IT Team immediately.
# Switch to backup servers within 10 minutes.
# Post announcements on social media to inform customers.
# Recover lost data using backup files.
# Document the problem and solution in the maintenance log.

'''Purpose:'''
This minimizes company losses and restores operations quickly.

==== 5. Emergency Contact List ====
* Operations Manager – 09123456789
* IT Support Team – 09987654321
* Security Officer – 09771234567

=== Audit Trail / Change Log ===
* Added backup schedule for customer data.
* Updated security procedures for admin accounts.
* Improved emergency response steps for faster recovery.

=== Conclusion ===
A proper Business Maintenance Plan helps the company avoid major losses, protect customer data, and maintain smooth business operations. Through regular maintenance, proper documentation, and emergency planning, the business can continue operating successfully even during unexpected situations.</text>
      <sha1>cyz5ym4byvh6sq2y9sxshl8dzpkiptv</sha1>
    </revision>
  </page>
  <page>
    <title>Economics/Introduction</title>
    <ns>0</ns>
    <id>8030</id>
    <revision>
      <id>2807819</id>
      <parentid>2247380</parentid>
      <timestamp>2026-05-06T15:12:58Z</timestamp>
      <contributor>
        <username>Jevil64</username>
        <id>2955984</id>
      </contributor>
      <minor />
      <comment>minor copyediting in later sections</comment>
      <origin>2807819</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4564" sha1="1k9841d3ksymxkaw23j63n9yzgxegrw" xml:space="preserve">{{economics}}

==The Basic Economic Problem==

&lt;i&gt;&quot;Economics is the study of the division of scarce resources between unlimited needs and wants&quot;&lt;/i&gt;

Let us explain further.    You are the head of your household.  There are several things which are needed to keep the household running, e.g. food, electricity, gas and so forth.  There are also wants, which may not have to be fulfilled per se, but will be acquired somehow.

A store manager has a limited budget to spend on various &lt;i&gt;wants&lt;/i&gt;, his resources are &lt;i&gt;scarce&lt;/i&gt;.  He could spend his money on providing more staff, bringing in more stock from a wholesaler, marketing his product to increase demand through advertising, decorating or improving his store and paying for essential services like water and electricity. Effectively he has unlimited &lt;i&gt;wants&lt;/i&gt; but limited means or resources.  We say that his resources are &lt;i&gt;scarce&lt;/i&gt;.  The store manager must &lt;i&gt;economise&lt;/i&gt; by choosing to use his resources to satisfy certain wants such as marketing instead of others like stock purchase.  As earlier stated, economics is the study of the division of these resources to best satisfy the unlimited wants.

Every economy in the world face three main basic economics problems because the needs and wants of the society are  unlimited but the resources available to satisfy those are limited. Whether a country is rich or poor this is a common situation to all of them. The main economics problem are:

# What to Produce in which quantities?
# How to Produce?
# For whom to Produce?

Let us look at the problems in detail.

=== What to produce ?===

In any society there are unlimited wants, but resources are limited, or &quot;scarce.&quot; Furthermore, these resources have alternative uses. Due to this, each society has to decide what they are to produce using these scarce resources. As such, each economy has to make a choice by thinking what kinds of products or what quantity is to be produced. For example, an economy has to decide whether to produce more services such as transport or hospitals, consumable goods like more clothes and houses, or more capital goods such as roads, buildings, etc.
The economy must decide which goods and services to produce and which goods and services to exclude from production
is the problem of choice between commodities.
Note that the decision of what to produce is made, having the consumers preferences in mind.

=== How to produce? ===

 
The problem of ‘how to produce’ means which combination of resources is to be used for the production-of goods and which technology is to be made use of in production. Once the society has decided what goods and services are to be produced and in what quantities, it must then decide how these goods shall be produced. There are various alternative methods of producing a good and the economy has to choose among them.
For example, cloth can be produced either with automatic looms or power looms. Fields can be irrigated (and hence wheat can be produced) by building small irrigation works like tube-wells and tanks or by building large canals and dams. Therefore, the economy has to decide whether cloth is to be produced by power looms or automatic looms. Similarly, it has to decide if the irrigation is to be done by minor irrigation works or by major works. So obviously, it is a problem of the choice of production.

=== For whom to produce? ===

Once an economy has produced goods and services, it also has to decide who will consume those goods. This will be decided differently by the nature of the economy. 

Though the above three problems are common to each economy, an economy can take different approaches to solve them. Depending on the approach, economies are organised in different ways. This is why different economies exist, such as market economies, centrally planned economies, mixed economies, and so forth.

==Another Perspective==

Think about why you get up in the morning.

What motivates you to get out of bed?  Eat food?  Work?  Dance?  Make friends? Fear of starvation, failure or ridicule?

We are concerned with both the value of doing any given thing as well as the value of not doing any given thing (known hereafter as opportunity cost).  Economics comes down to one idea, happiness.  

Economics is the quantitative and eventual qualitative description of every human activity in our world. By studying economics you are analysing rational behavior to a higher degree than the level that we live by in our daily lives.

==See also==
* [[School:Economics]]

[[Category:Economics]]</text>
      <sha1>1k9841d3ksymxkaw23j63n9yzgxegrw</sha1>
    </revision>
  </page>
  <page>
    <title>SPIR608 Political Simulations and Gaming/2011/Week 6</title>
    <ns>0</ns>
    <id>107053</id>
    <revision>
      <id>2807858</id>
      <parentid>2197209</parentid>
      <timestamp>2026-05-07T06:50:59Z</timestamp>
      <contributor>
        <username>CommonsDelinker</username>
        <id>9184</id>
      </contributor>
      <comment>Replacing Gen-commons.jpg with [[File:1954_Geneva_Conference.jpg]] (by [[:c:User:CommonsDelinker|CommonsDelinker]] because: [[:c:COM:FR|File renamed]]: [[:c:COM:FR#FR2|Criterion 2]] (meaningless or ambiguous name)).</comment>
      <origin>2807858</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4700" sha1="bqena8hwcjchgtwnqbotpqucqa7se2l" xml:space="preserve">{{Template:SPIR608 Political Simulation and Gaming/2011/nav}}
Friday 25th February

*&lt;big&gt;'''Week 6 Discussion of Serious Games, Vietnam 1955''.&lt;/big&gt;
{{Template:Gaming card
|card_type ='''Game'''
|card_name =Vietnam 1955:&lt;br&gt;
|card_centijimbos =10

|background_color=LightSalmon
|caption_background_color=lemonchiffon
|outside_background_color=DarkRed

|image_name =[[File:1954 Geneva Conference.jpg|180px]]
|image_caption =Geneva Peace Conference 1954
|image_credit =US Army Photograph

|body_title =Russell King
|body_text =Serious Games&lt;br&gt;Role playing game leading up to the [[W:Geneva Conference (1954)|Geneva Conference 1954]]
|body_link =
}}

'''* Is the design of the game's mechanics (board, pieces, cards, etc.) fit for purpose?'''
The game was comprised of just an introductory booklet and A4 sheets with victory points. There was no board or pieces. Perhaps it would have been easier to play if poker chips were used to keep track of the game score. A map of 1950s Vietnam would have helped. The lack of a rule book was a problem at some points in the game. ''Vietnam 1955'' was more of a interactive simulation than a game. It seemed more like reality as the rules could be changed to fit the progress of the game. However, if we played it again, we could learn how to&quot;game the game&quot;. It was unclear whether the way to win ''Vietnam 1955'' was gaining more victory points or acheiving certain goals. Heading at top of A4 sheets with victory points made people think they needed to fufill the conditions at the top. If you wanted to simulate history more closely, people should be forced to play more in character.


'''* Is the game enjoyable and sociable to play?'''
We loved it! ''Vietnam 1955'' was even better than ''Comrade Koba''.

'''* What techniques does the game use to model its chosen subject?'''
''Vietnam 1955'' was uses the [[W:Free kriegsspiel| free kriegsspiel]] system. Perhaps the game is too free sometimes. Umpire was invigilating and interacting with players in order to manipulate the game in certain directions where necessary. It showed that Russell is very experienced at running from his work with the NHS. The umpire having so much poweer can sometimes seem to be unfair. But this does work as a technique to make the simulation more realistic. ''Vietnam 1955'' works by using the living labour of an umpire rather than the [[w:dead labour|dead labour]] of a designer &quot;congealed&quot; in a board game.

'''* How does the game combine abstraction and realism in its workings?'''
The biggest problem was that choices do not have real-life consequences so that threatening a nuclear war was an acceptable risk in the game. It was realistic that we did not know what the other players' victory conditions were. ''Vietnam 1955'' was historically realistic in how the different players divided into the two Cold War blocs during the game.

'''* How accurately does the game simulate the decision-making processes faced by the real-life protagonists of its chosen subject?'''
''Vietnam 1955'' is a bit like ''Origins of World War II''. However it did seem strange that the French Indo-China and Vietminh players could work together so easily. Victory points do ensure that there are areas where people must confront each other, and other areas where they need to co-operate. As a participatory event, ''Vietnam 1955'' was a bit like [[W:Rousseau|Rousseau]]'s theory of the festival.

'''* What political lessons can people learn by playing the game?'''
''Vietnam 1955'' teaches its players how diplomacy works and the need to create alliances. You need to know how to protect your own interests and predict what other people are striving for. The umpire should have deducted victory points when players were breaking out of their roles.

'''* How would you improve the structure and mechanics of the game?'''
* Include name badges and country flags in game equipment;
* Have the players in separate rooms so they would only meet when they want to meet;
* Vary victory conditions if game played again;
* Losing victory points for breaking out of role.
* Have a giant map of Vietnam to orientate the players geographically;
* Use chips to keep track of gaining for losing victory points;
* Make the room layout more specific to the game, such as having negotiating areas;
* Have a player representing the world media who issues press releases and news bulletins to give the game more flavour.

==Links==
* [http://www.facebook.com/album.php?aid=331729&amp;id=549261319&amp;l=e26b903997 Photos of Week 6 game playing]
* [[SPIR608_Political_Simulation_and_Gaming/2011/Vietnam 1955 Scenario |Vietnam 1955 Scenario]]
* [http://www.wargamedevelopments.org/cow.htm Wargame Developments]</text>
      <sha1>bqena8hwcjchgtwnqbotpqucqa7se2l</sha1>
    </revision>
  </page>
  <page>
    <title>SPIR608/Games</title>
    <ns>0</ns>
    <id>107710</id>
    <revision>
      <id>2807859</id>
      <parentid>2730744</parentid>
      <timestamp>2026-05-07T06:51:00Z</timestamp>
      <contributor>
        <username>CommonsDelinker</username>
        <id>9184</id>
      </contributor>
      <comment>Replacing Gen-commons.jpg with [[File:1954_Geneva_Conference.jpg]] (by [[:c:User:CommonsDelinker|CommonsDelinker]] because: [[:c:COM:FR|File renamed]]: [[:c:COM:FR#FR2|Criterion 2]] (meaningless or ambiguous name)).</comment>
      <origin>2807859</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5966" sha1="djs9hptoiab5x22571ncnw3vz30hubf" xml:space="preserve">{{Template:SPIR608 Political Simulations and Gaming/2011/nav}}
==Games on course==
{{Template:Gaming card
|card_type ='''Game'''
|card_name =[[W:Monopoly (game)|Monopoly]]:&lt;br&gt;
|card_centijimbos =10

|background_color=LightSalmon
|caption_background_color=lemonchiffon
|outside_background_color=DarkRed

|image_name =[[File:Landlords Game board based on 1924 patent.png|100px]]
|image_caption =[[W:Landlords Game|Landlords Game]] board based on 1924 patent, precursor of Monopoly
|image_credit =

|body_title =[[W:Elizabeth Magie|Elizabeth Magie]] and [[W:Charles Darrow|Charles Darrow]]
|body_text =[[W:Parker Brothers|Parker Brothers]], 1934
|body_link =
}}{{Template:Gaming card
|card_type ='''Game'''
|card_name =[[W:Le Jeu de la Guerre|Game of War]]:&lt;br&gt;
|card_centijimbos =10

|background_color=LightSalmon
|caption_background_color=lemonchiffon
|outside_background_color=DarkRed

|image_name =[[File:Game of War, Brazil.jpg|180px]]
|image_caption =Guy Debord's Game of War being played in Belo Horizonte, Brazil
|image_credit =Richard Barbrook

|body_title =[[W:Guy Debord|Guy Debord]] and &lt;br&gt;[[W:Alice Becker-Ho|Alice Becker-Ho]]
|body_text =Published as ''Le Jeu de la Guerre'', 1987
|body_link =
}}{{Template:Gaming card
|card_type ='''Game'''
|card_name =[[W:1776 (game)|1776]]:&lt;br&gt;
|card_centijimbos =10

|background_color=LightSalmon
|caption_background_color=lemonchiffon
|outside_background_color=DarkRed

|image_name =[[File:Washington Crossing the Delaware by Emanuel Leutze, MMA-NYC, 1851.jpg|180px]]
|image_caption =''[[W:George Washington|Washington]] Crossing the [[W:Delaware River|Delaware]]'' by [[W:Emanuel Leutze|Emanuel Leutze]]
|image_credit =

|body_title =Randell Reed
|body_text =[[W:Avalon Hill|Avalon Hill]], 1974&lt;br&gt;''The Game of the [[W:American Revolution|Revolutionary War]]''
|body_link =
}}{{Template:Gaming card
|card_type ='''Game'''
|card_name =[[W:Kingmaker (board game)|Kingmaker]]:&lt;br&gt;
|card_centijimbos =10

|background_color=LightSalmon
|caption_background_color=lemonchiffon
|outside_background_color=DarkRed

|image_name =[[File:Battle of Barnet retouched.jpg|80px]]
|image_caption =The death of [[W:Richard Neville, 16th Earl of Warwick|Warwick]] at the [[W:Battle of Barnet|Battle of Barnet]]
|image_credit =

|body_title =Andrew McNeil
|body_text =PhilMar, 1974&lt;br&gt;Game of the English [[W:Wars of the Roses|Wars of the Roses]]
|body_link =
}}{{Template:Gaming card
|card_type ='''Game'''
|card_name =[[W:War on Terror (game)|The War on Terror]]:&lt;br&gt;
|card_centijimbos =10

|background_color=LightSalmon
|caption_background_color=lemonchiffon
|outside_background_color=DarkRed

|image_name =[[File:War on Terror montage1.png|140px]]
|image_caption =
|image_credit =

|body_title =Andy Tompkins and Andrew Sheerin
|body_text =[[W:TerrorBull Games|TerrorBull Games]], 2006&lt;br&gt;Satirical game of the [[W:War on Terror|War on Terror]]
|body_link =
}}{{Template:Gaming card
|card_type ='''Game'''
|card_name =[[W:Origins of World War II (game)|Origins of&lt;br&gt; World War II]]:&lt;br&gt;
|card_centijimbos =10

|background_color=LightSalmon
|caption_background_color=lemonchiffon
|outside_background_color=DarkRed

|image_name =[[File:Munich Agreement Bundesarchiv Bild 183-R69173.jpg|180px]]
|image_caption =Assembled heads of state in Munich, 29 September 1938
|image_credit =Deutsches Bundesarchiv

|body_title =[[W:Jim Dunnigan|Jim Dunnigan]]
|body_text =[[W:Avalon Hill|Avalon Hill]], 1971&lt;br&gt;Based on [[W:AJP Taylor|A. J. P. Taylor]]'s view of the origins of [[W:World War II|World War II]]
|body_link =
}}{{Template:Gaming card
|card_type ='''Game'''
|card_name =Modern Society:&lt;br&gt;
|card_centijimbos =10

|background_color=LightSalmon
|caption_background_color=lemonchiffon
|outside_background_color=DarkRed

|image_name =[[File:Modern Society.jpg|180px]]
|image_caption =East End cityscape
|image_credit =Fabian Tompsett

|body_title =Jussi Autio
|body_text =
|body_link =
}}{{Template:Gaming card
|card_type ='''Game'''
|card_name =Vietnam 1955:&lt;br&gt;
|card_centijimbos =10

|background_color=LightSalmon
|caption_background_color=lemonchiffon
|outside_background_color=DarkRed

|image_name =[[File:1954 Geneva Conference.jpg|180px]]
|image_caption =Geneva Peace Conference 1954
|image_credit =US Army Photograph

|body_title =Russell King
|body_text =Serious Games&lt;br&gt;Role playing game leading up to the [[W:Geneva Conference (1954)|Geneva Conference 1954]]
|body_link =
}}{{Template:Gaming card
|card_type ='''Game'''
|card_name =''Red Guard!'':&lt;br&gt;
|card_centijimbos =10

|background_color=LightSalmon
|caption_background_color=lemonchiffon
|outside_background_color=DarkRed

|image_name =
|image_caption =
|image_credit =

|body_title =Brian Train
|body_text =Game of the Chinese [[W:Cultural Revolution|Cultural Revolution]]
|body_link =
}}
{{Template:Gaming card
|card_type ='''Game'''
|card_name =''Liberté'':&lt;br&gt;
|card_centijimbos =10

|background_color=LightSalmon
|caption_background_color=lemonchiffon
|outside_background_color=DarkRed

|image_name =[[File:Liberté at SPIR608 March 2011.jpg|180px]]
|image_caption =''Liberté'' in play, University of Westminster
|image_credit =Fabian Tompsett

|body_title =[[W:Martin Wallace (game designer)|Martin Wallace]]
|body_text =Electoral game of the [[W:French Revolution|French Revolution]]&lt;br&gt;
|body_link =
}}
{{-}}
==Other Wikiversity games==
{{Template:Gaming card
|card_type ='''Game'''
|card_name =[[Forbidden Kingdom (Strategic Simulation)|Forbidden Kingdom]]:&lt;br&gt;
|card_centijimbos =10

|background_color=LightSalmon
|caption_background_color=lemonchiffon
|outside_background_color=DarkRed

|image_name =[[File:Forbidden Kingdom Sim.jpg|180px]]
|image_caption =Forbidden Kingdom board ready for play
|image_credit =[[User:Pnoble805]]

|body_title =[[User:Pnoble805]]
|body_text = Wikiversity 2008&lt;br&gt;Forbidden Kingdom is a simulation of decision making quandries that might be faced in the Communist Party in the People's Republic of China.
|body_link =
}}</text>
      <sha1>djs9hptoiab5x22571ncnw3vz30hubf</sha1>
    </revision>
  </page>
  <page>
    <title>Swahili I/Numbers and Time</title>
    <ns>0</ns>
    <id>109135</id>
    <revision>
      <id>2807800</id>
      <parentid>2741078</parentid>
      <timestamp>2026-05-06T12:06:49Z</timestamp>
      <contributor>
        <username>~2026-27543-11</username>
        <id>3070700</id>
      </contributor>
      <origin>2807800</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3302" sha1="muqpdyw7o4htln1ux9yy9b4htx3npsv" xml:space="preserve">== Numbers ==
=== 1-10 ===
1 = Moja (adj. -moja)&lt;br&gt;
2 = Mbili (adj. -wili) &lt;br&gt;
3 = Tatu (adj. -tatu) &lt;br&gt;
4 = Nne (adj. -nne) &lt;br&gt;
5 = Tano (adj. -tano) &lt;br&gt;
6 = Sita &lt;br&gt;
7 = Saba &lt;br&gt;
8 = Nane &lt;br&gt;
9 = Tisa/Kenda&lt;br&gt; Kenda is used in a different dialect mostly from Burundi, Rwanda and D.R.C but Tisa is used by Tanzania and Kenya more often. 

10 = Kumi &lt;br&gt;

=== 10-20 ===
11 = Kumi na moja (Na = And -&gt; Kumi na moja = Ten and One) &lt;br&gt;
12 = Kumi na mbili &lt;br&gt;
13 = Kumi na tatu &lt;br&gt;
...
20 = Ishirini &lt;br&gt;
(21 = Ishirini na moja... ) &lt;br&gt;
=== Tens ===
30 = Thelathini &lt;br&gt;
40 = Arobaini &lt;br&gt;
50 = Hamsini &lt;br&gt;
60 = Sitini &lt;br&gt;
70 = Sabini &lt;br&gt;
80 = Themanini &lt;br&gt;
90 = Tisini &lt;br&gt;

=== 100+ ===
100 = Mia moja &lt;br&gt;
(101 = Mia moja na moja... ) &lt;br&gt;
200 = Mia mbili &lt;br&gt;
...
1,000 = Elfu

2000 = elfu mbili
== Time ==
The swahili time is expressed very differently from standard time in other parts of the world. Instead of midnight and noon, Swahili time is based on sunset and sunrise. As most Swahili-speaking countries are located near the equator, sunset and sunrise are mostly constant year-round, and defined to be 6:00 P.M. and 6:00 A.M., respectively in standard time. Hence, 6:00 A.M. is the zero hour (0:00 or 12:00) in Swahili time, 7:00 A.M. is the first hour of the day (1:00 in the Morning) and so on. An easy translation to Swahili time is to subtract six hours from the standard clock time, which is how many natives adjust. For example, 11:30 A.M. in standard time is 5:30 in the morning in Swahili time. Instead of A.M. and P.M., Swahili time expresses the hour followed by the portion of the day. &lt;br&gt;
* ''Alfajiri'' = early morning, before the sun has fully risen &lt;br&gt;
* ''Asubuhi'' = morning, roughly between sunrise and noon &lt;br&gt;
* ''Mchana'' = daytime, between sunrise 6:A.M. and sunset 6:00 P.M. &lt;br&gt;
* ''Jioni'' = evening, between sunset 6:P.M. and sunrise 6:00 A.M. &lt;br&gt;
* ''Usiku'' = night time, from sunset until early morning again &lt;br&gt;

As with standard time, the hour (''saa'') is expressed first, followed by the minutes (''dakika''). There are also abbreviations for half past (''nusu'', half) and others (''kasorobo'', less a quarter).  Here are some examples of time-telling to help you understand how it is done. 
{| class=&quot;wikitable&quot;
|-
! Standard Time !! Swahili Time !! Translation !! Note
|-
| 12:00 A.M.|| 6:00 at night || ''saa sita usiku'' || literally hour six night
|-
| 3:00 P.M.|| 9:00 in the afternoon || ''saa tisa mchana'' ||
|-
| 7:30 P.M.|| 1:30 in the evening || ''saa moja na nusu jioni'' || literally hour one and a half evening
|-
| 1:05 P.M.|| 7:05 in the afternoon || ''saa saba na dakika tano mchana'' || literally hour seven and minutes five afternoon
|-
| A quarter to 8:00 A.M.(7.45 A.M.)|| A quarter to 2:00 in the morning || ''saa mbili kasorobo asubuhi'' || literally hour two less one quarter morning
|-
| Almost 11:00 A.M.|| Almost 5:00 in the morning || ''saa tano kasoro asubuhi. (Karibu saa tano)'' || literally hour five less morning (Literally almost hour five)
|}

== Advice ==
You should make flashcards for these with the actual digits on one side and the Swahili word on the other and quiz yourself, looking at both sides and saying, aloud, what is on the other. 
=== Exercises ===
[[/Exercises/]]

[[Category:Swahili]]</text>
      <sha1>muqpdyw7o4htln1ux9yy9b4htx3npsv</sha1>
    </revision>
  </page>
  <page>
    <title>The necessities in Digital Design</title>
    <ns>0</ns>
    <id>119422</id>
    <revision>
      <id>2807824</id>
      <parentid>2807363</parentid>
      <timestamp>2026-05-06T17:37:01Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/*  Timing Analysis  */</comment>
      <origin>2807824</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9835" sha1="as70vuk2xj6ulczb5d3z784n7k0w8fh" xml:space="preserve">
== ''' Number Systems '''== 

=== ''' Binary Representation '''=== 
* Binary Numbers ([[Media:DD1.1.A.BinaryNum.20130918.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD1.2.A.HexaNum.20130918.pdf|A.pdf]])
* Other Codes ([[Media:DD1.3A.Code.20250329.pdf|A.pdf]])

=== ''' Binary Arithmetic '''=== 
* Binary Arithmetic ([[Media:DD1.4.A.BinaryArith.20150425.pdf|A.pdf]])
* BCD Arithmetic ([[Media:DD1.5.A.BCDArith.20130918.pdf|A.pdf]])

=== ''' C Program Examples '''=== 
* Binary Numbers in C programs ([[Media:DD1.6.A.BNumInC.20140103.pdf|A.pdf]])
* Binary Addition in C programs ([[Media:DD1.7.A.BArithInC.20140103.pdf|A.pdf]])
&lt;/br&gt;

* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|C.pdf]])
&lt;/br&gt;

=== ''' Floating Point Numbers '''=== 

* Floating Point Representations  ([[Media:CDesign.5.A.FPoint.20140121.pdf|5A.pdf]])&lt;/br&gt;
:: See [http://www.iro.umontreal.ca/~aboulham/F1214/Session%206Arithm/Floating_Point_Numbers.pdf Floating Point Overview]
:: See [http://www.cs.auckland.ac.nz/~patrice/210-2006/210%20LN04_2.pdf Offset Binary Overview]
:: See [http://www.intersil.com/content/dam/Intersil/documents/an96/an9657.pdf Offset Binary &amp; Sin / Cosine]
:: See [http://www.ee.ic.ac.uk/hp/staff/dmb/courses/dig2/4_Analog.pdf Offset Binary &amp; ADC / DAC]
&lt;/br&gt;

=== ''' Interfacing Digital and Analog Signals '''=== 
* Sampling and Quantization   ([[Media:DD1.10.A.SampleQuant.20150425.pdf|A.pdf]])
* Digital-to-Analog Conversion  ([[Media:DD1.8.A.DAC.20140208.pdf|A.pdf]])
* Analog-to-Digital Conversion ([[Media:DD1.9.A.DAC.20140208.pdf|A.pdf]])
&lt;/br&gt;

== '''Combinational Circuits'''==

=== ''' Design '''=== 
* Boolean Algebra ([[Media:DD2.A1.BAlgebra.20250503.pdf|A1.pdf]])
* Truth Tables ([[Media:DD2.A2.TTable.20250424.pdf|A2.pdf]])
* K-Map ([[Media:DD2.A3.KMap.20250424.pdf|A3.pdf]])
* Design Examples ([[Media:DD2.A4.CombEx.20250414.pdf|A4.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Decoder ([[Media:DD2.B.1.Decoder.20130928.pdf|B1.pdf]])
* Encoder ([[Media:DD2.B.2.Encoder.20130917.pdf|B2.pdf]])
* Multiplexer ([[Media:DD2.B.3.Multiplexer.20130928.pdf|B3.pdf]])
* Adder ([[Media:DD2.B.4..Adder.20131007.pdf|B4.pdf]], [[Media:Fa.sch.20131002.pdf|fa.sch.pdf]], [[Media:Adder4.sch.20131002.pdf|adder4.sch.pdf]])
&lt;/br&gt;

=== ''' Design Metric '''=== 
* Noise Margin ([[Media:DD2.C1.NoiseMargin.20250415.pdf|C1.pdf]])
&lt;/br&gt;

== '''Sequential Circuits'''==

=== ''' Design '''=== 
* Types of Flip-Flops ([[Media:CDesign.1.A.FF.20130412.pdf |1A.pdf]])&lt;/br&gt;
* Latches and Flipflops ([[Media:DD3.A.1.LatchFF.20160308.pdf|A1.pdf]])
* State Transition Table ([[Media:DD3.A.2.pdf|A2.pdf]])
* FSM (Finite State Machine) ([[Media:DD3.A.3.FSM.20131030.pdf|A3.pdf]])
&lt;/br&gt;

* The Classic FF Design ([[Media:DD3.A.6.ClassicFF.20131126.pdf|A7.pdf]])
* The Modern FF Design ([[Media:DD3.A.6.ClassicFF.20131204.2.pdf|A8.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Latches and Flip-flops ([[Media:DD3.B.1.LatchFF.20131008.pdf|B1.pdf]])
* Registers ([[Media:DD3.B.2.Register.20150326.pdf|B2.pdf]], [[Media:Register.20131118.pdf|register.pdf]])
* Counters ([[Media:DD3.B.2.Counter.20150420.pdf|B3.pdf]])
&lt;/br&gt;

=== ''' Timing Analysis '''=== 
* Metastability ([[Media:DD3.A.4.MetaState.20131030.pdf|A4.pdf]])
* Flip-flop Timing ([[Media:DD3.A5.FFTiming.20260504.pdf|A5.pdf]])
* SR Latch Forbidden State ([[Media:DD3.A.5.ForbiddenState.20131030.pdf|A6.pdf]])
&lt;/br&gt;

* FF Min Max Timing Constraints ([[Media:CArch.MinMaxTiming.20131121.pdf |pdf]])
* FF Clock Skew Timing Constraints ([[Media:CArch.ClockSkew.20131121.pdf |pdf]])

* Synchronizer ([[Media:CArch.Synchronizer.20131216.pdf |pdf]])
* Resolution Time Analysis  ([[Media:CArch.Resolution.20131216.pdf |pdf]])
&lt;/br&gt;

== '''Finite State Machine'''==

* FSM State Encoding 
* FSM Types : Mealy and Moore Machines
* FSM Example ([[Media:CArch.2.A.FSMExample.20141018.pdf |pdf]])
&lt;/br&gt;

== '''Array  Devices''' ==

=== ''' Memory Arrays '''=== 
* RAM 
** RAM Structure ([[Media:DD4.A.1.RAM.20131111.pdf|A.pdf]])
** RAM Timing ([[Media:DD4.B.1.RAMTiming.20131130.pdf|B.pdf]])
** FPGA RAM ([[Media:DD4.C.1.FPGARAM.20160513.pdf|C.pdf]])

* ROM
&lt;/br&gt;

=== ''' Logic Arrays '''=== 
* PLA 
* PAL
* PLD
* FPGA
** FPGA Structure
** FPGA Configuration ([[Media:DD4.C.1.FPGAConf.20131130.pdf|B.pdf]])
&lt;/br&gt;
&lt;/br&gt; 
[http://www.ece.cmu.edu/~ece548/localcpy/sramop.pdf  Synchronous SRAM Timing] &lt;/br&gt;
[http://www.micron.com/~/media/Documents/Products/Technical%20Note/DRAM/tn4529.pdf Asynchronous SRAM Timing]&lt;/br&gt;
[http://www.ece.cmu.edu/~ece548/localcpy/dramop.pdf DRAM Timing] &lt;/br&gt;

[http://www.ece.unm.edu/~jimp/415/slides/fpga_arch1.pdf FPGA Architectures] &lt;/br&gt;
[http://www.engr.siu.edu/~haibo/ece428/notes/ece428_fpgaarch.pdf CPLD &amp; FPGA] &lt;/br&gt;
&lt;/br&gt;

== ''' RTL Design Techniques''' ==
&lt;/br&gt;

''' Design Methodology '''
&lt;/br&gt;


''' Synthesis '''
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

== '''Logic Families and IOs''' ==
* BJT Based 
:: DTL (Diode-Transistor Logic)
:: TTL (Transistor-Transistor Logic)
:: ECL (Emitter-Coupled Logic)
 
* MOS Based
:: CMOS (Complementary MOS)
:: Pseudo-nMOS 
:: Transmission Gate
:: BiCMOS (Bipolr + CMOS)
 
* Dynamic CMOS 
:: Domino
:: Clocked-CMOS (C&lt;sup&gt;2&lt;/sup&gt;MOS)
&lt;/br&gt; 

* Modern I/O Standards
:: TTL and LVTTL (Low Voltage TTL)
:: CMOS and LVCMOS (Low Voltage CMOS)
:: SSTL (Stub Series Terminated Logic)
:: HSTL (High Speed Tranceiver Logic)
:: LVDS (Low Voltage Differential Signaling) 

&lt;/br&gt;

* Wikipedia Pages for Logic Families ([[Media:Logic Families.wiki.20140812.pdf|A.pdf]])
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&lt;[[The necessities in Computer Design]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Architecture]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Organization]]&gt;  &lt;/br&gt;


&lt;/br&gt; &lt;/br&gt;

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

== '''Old''' ==
'''Until 2011.12''' 
'''Chapter 1. Binary Numbers'''
* 1.1 Binary Numbers([[Media:BinaryNumbers.1.A.pdf|pdf]])

''' Minterm, Maxterm, HW '''
* 1.1 Lecture01([[Media:DigitalDesign.20110922.pdf|pdf]])

''' Overflow HW '''
* Overflow Table([[Media:Overflow table.20110924.pdf|pdf]])

''' K-Map '''
* K-Map([[Media:DigitalDesign.20110926.pdf|pdf]])

''' Binary Adder '''
* Binary Adder (C, S) ([[Media:DigitalDesign.20110929.pdf|pdf]])
* Overflow detection circuit (V) ([[Media:HW Overflow20111001.pdf|pdf]])

''' BCD to Ex3 Code Coversion, Dont' Care '''
* BCD to Ex3 Code Conversion ([[Media:DigitalDesign.20111006.pdf|pdf]])

''' Prime Implicant, Dont' Care '''
* Prime Implicant, Don't Care ([[Media:DigitalDesign.20111010.pdf|pdf]])
* HW 3.6 - explain the method of combining 0's and X's

''' Multiplexer / Demultiplexer '''
* Multiplexer ([[Media:DigitalDesign.20111024.pdf|pdf]])
* HW (TBD)

''' Flip Flop / Latch '''
* FF &amp; Latch ([[Media:DigitalDesign.20111027.pdf|pdf]])
* FF &amp; Latch HW ([[Media:DigitalDesign (HW).20111027.pdf|pdf]])
* Gated D Latch &amp; Master-Slave D FlipFlop ([[Media:DigitalDesign.20111031.pdf|pdf]])
* HW (Forbidden state and Indeterminate state) ([[Media:DigitalDesign (HW).20111102.pdf|pdf]]) (note in #2, S' R' instead of S R)
* Classical Edge Triggered D FlipFlop ([[Media:DigitalDesign.20111112.pdf|pdf]])
* HW (addition in SW and HW) ([[Media:DigitalDesign (HW).20111112.pdf|pdf]]) 
* FSM1  ([[Media:DigitalDesign.FSM1.20111117.pdf|pdf]]) 
* FSM2  ([[Media:DigitalDesign.FSM2.20111117.pdf|pdf]]) 
* HW (FSM Waveforms) ([[Media:DigitalDesign (HW).20111118.pdf|pdf]]) 
 
 
''' Counter '''
* Sychronous Counter ([[Media:DigitalDesign.20111121.pdf|pdf]])
* Ripple Counter, Multiplexer, Tri-state buffer([[Media:DigitalDesign.20111124.pdf|pdf]])
* Register ([[Media:DigitalDesign.register.20111201.pdf|pdf]])
* Timing ([[Media:DigitalDesign.timing.20111201.pdf|pdf]])
* HW (Multiplexer, Shift Register) ([[Media:DigitalDesign (HW).20111201.pdf|pdf]])
* Universal Shift Register, Memory Cell ([[Media:DigitalDesign.20111206.pdf|pdf]])
* HW (Bit Serial Adder) ([[Media:DigitalDesign (HW).20111206.pdf|pdf]])
 
 
''' Memory '''
* Memory ([[Media:DigitalDesign.20111208.pdf|pdf]])

''' Comparator, Multiplier '''
* Comparator, Multiplier ([[Media:DigitalDesign.20111219.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111219.draw.pdf|2.pdf]])
 
 
'''Multiplexer based design method '''
* Multiplexer Design Method ([[Media:DigitalDesign.20111221.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111221.draw.pdf|2.pdf]])


midterm result ([[Media:MidReult.20111027.pdf|pdf]])

 
* Edge Triggered Flip Flop ([[Media:EdgeTrigFF.20111224.pdf|pdf]])
* FF Timing ([[Media:FFTiming.20111203.pdf|pdf]])


&lt;/br&gt; &lt;/br&gt;
'''Until 2013.07''' 

''' Number Systems '''
* Binary Numbers ([[Media:DD.1.A.BinNum.20130309.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD.1.B.HexaNum.20130417.pdf|B.pdf]])
* Numbers in C programs ([[Media:DD.1.C.CNum.20130309.pdf|C.pdf]])
* Codes ([[Media:DD.1.D.Coding.20130319.pdf|pdf]])
&lt;/br&gt;

&lt;/br&gt;
* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|pdf]])
&lt;/br&gt;

''' Combinational Circuits '''
* Truth Tables and Boolean Functions ([[Media:DD.2.A.TTable.20130325.pdf|2A.pdf]])&lt;/br&gt;
* K-Map ([[Media:DD.2.A.KMap.20130329.pdf|2B.pdf]])&lt;/br&gt;
* Binary Addition in C ([[Media:DD.2.C.BAinC.20130329.pdf|2.C.pdf]])&lt;/br&gt;
* Binary Arithmetic ([[Media:DD.2.D.BAri.2013.pdf|2.D.pdf]])&lt;/br&gt;
* Boolean Algebra ([[Media:DD.2.E.BAlgebra.20130419.pdf|2.E.pdf]])&lt;/br&gt;
&lt;/br&gt;

''' Sequential Circuits '''
* Latches and Flip-flops ([[Media:DD.3.A.LatchFF.20130413.pdf|3A.pdf]])&lt;/br&gt;
* FSM (Finite State Machine) ([[Media:DD.3.B.FSM.20130417.pdf|3B.pdf]])&lt;/br&gt;
* SR Latch Forbidden State ([[Media:DD.3.C.FState.20130413.pdf|3C.pdf]])&lt;/br&gt;
* Flip-flop Timing ([[Media:DD.3.D.Timing.20130413.pdf|3D.pdf]])&lt;/br&gt;
* Metastability ([[Media:DD.3.E.MetaState.20130628.pdf|3E.pdf]])&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&quot;[[The necessities in Computer Design]]&quot;  &lt;/br&gt;
&quot;[[The necessities in Computer Architecture]]&quot;  &lt;/br&gt;

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>as70vuk2xj6ulczb5d3z784n7k0w8fh</sha1>
    </revision>
    <revision>
      <id>2807826</id>
      <parentid>2807824</parentid>
      <timestamp>2026-05-06T17:38:20Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/*  Timing Analysis  */</comment>
      <origin>2807826</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9835" sha1="7ponhn49dnu90bspopx8uyhowm544fm" xml:space="preserve">
== ''' Number Systems '''== 

=== ''' Binary Representation '''=== 
* Binary Numbers ([[Media:DD1.1.A.BinaryNum.20130918.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD1.2.A.HexaNum.20130918.pdf|A.pdf]])
* Other Codes ([[Media:DD1.3A.Code.20250329.pdf|A.pdf]])

=== ''' Binary Arithmetic '''=== 
* Binary Arithmetic ([[Media:DD1.4.A.BinaryArith.20150425.pdf|A.pdf]])
* BCD Arithmetic ([[Media:DD1.5.A.BCDArith.20130918.pdf|A.pdf]])

=== ''' C Program Examples '''=== 
* Binary Numbers in C programs ([[Media:DD1.6.A.BNumInC.20140103.pdf|A.pdf]])
* Binary Addition in C programs ([[Media:DD1.7.A.BArithInC.20140103.pdf|A.pdf]])
&lt;/br&gt;

* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|C.pdf]])
&lt;/br&gt;

=== ''' Floating Point Numbers '''=== 

* Floating Point Representations  ([[Media:CDesign.5.A.FPoint.20140121.pdf|5A.pdf]])&lt;/br&gt;
:: See [http://www.iro.umontreal.ca/~aboulham/F1214/Session%206Arithm/Floating_Point_Numbers.pdf Floating Point Overview]
:: See [http://www.cs.auckland.ac.nz/~patrice/210-2006/210%20LN04_2.pdf Offset Binary Overview]
:: See [http://www.intersil.com/content/dam/Intersil/documents/an96/an9657.pdf Offset Binary &amp; Sin / Cosine]
:: See [http://www.ee.ic.ac.uk/hp/staff/dmb/courses/dig2/4_Analog.pdf Offset Binary &amp; ADC / DAC]
&lt;/br&gt;

=== ''' Interfacing Digital and Analog Signals '''=== 
* Sampling and Quantization   ([[Media:DD1.10.A.SampleQuant.20150425.pdf|A.pdf]])
* Digital-to-Analog Conversion  ([[Media:DD1.8.A.DAC.20140208.pdf|A.pdf]])
* Analog-to-Digital Conversion ([[Media:DD1.9.A.DAC.20140208.pdf|A.pdf]])
&lt;/br&gt;

== '''Combinational Circuits'''==

=== ''' Design '''=== 
* Boolean Algebra ([[Media:DD2.A1.BAlgebra.20250503.pdf|A1.pdf]])
* Truth Tables ([[Media:DD2.A2.TTable.20250424.pdf|A2.pdf]])
* K-Map ([[Media:DD2.A3.KMap.20250424.pdf|A3.pdf]])
* Design Examples ([[Media:DD2.A4.CombEx.20250414.pdf|A4.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Decoder ([[Media:DD2.B.1.Decoder.20130928.pdf|B1.pdf]])
* Encoder ([[Media:DD2.B.2.Encoder.20130917.pdf|B2.pdf]])
* Multiplexer ([[Media:DD2.B.3.Multiplexer.20130928.pdf|B3.pdf]])
* Adder ([[Media:DD2.B.4..Adder.20131007.pdf|B4.pdf]], [[Media:Fa.sch.20131002.pdf|fa.sch.pdf]], [[Media:Adder4.sch.20131002.pdf|adder4.sch.pdf]])
&lt;/br&gt;

=== ''' Design Metric '''=== 
* Noise Margin ([[Media:DD2.C1.NoiseMargin.20250415.pdf|C1.pdf]])
&lt;/br&gt;

== '''Sequential Circuits'''==

=== ''' Design '''=== 
* Types of Flip-Flops ([[Media:CDesign.1.A.FF.20130412.pdf |1A.pdf]])&lt;/br&gt;
* Latches and Flipflops ([[Media:DD3.A.1.LatchFF.20160308.pdf|A1.pdf]])
* State Transition Table ([[Media:DD3.A.2.pdf|A2.pdf]])
* FSM (Finite State Machine) ([[Media:DD3.A.3.FSM.20131030.pdf|A3.pdf]])
&lt;/br&gt;

* The Classic FF Design ([[Media:DD3.A.6.ClassicFF.20131126.pdf|A7.pdf]])
* The Modern FF Design ([[Media:DD3.A.6.ClassicFF.20131204.2.pdf|A8.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Latches and Flip-flops ([[Media:DD3.B.1.LatchFF.20131008.pdf|B1.pdf]])
* Registers ([[Media:DD3.B.2.Register.20150326.pdf|B2.pdf]], [[Media:Register.20131118.pdf|register.pdf]])
* Counters ([[Media:DD3.B.2.Counter.20150420.pdf|B3.pdf]])
&lt;/br&gt;

=== ''' Timing Analysis '''=== 
* Metastability ([[Media:DD3.A.4.MetaState.20131030.pdf|A4.pdf]])
* Flip-flop Timing ([[Media:DD3.A5.FFTiming.20260505.pdf|A5.pdf]])
* SR Latch Forbidden State ([[Media:DD3.A.5.ForbiddenState.20131030.pdf|A6.pdf]])
&lt;/br&gt;

* FF Min Max Timing Constraints ([[Media:CArch.MinMaxTiming.20131121.pdf |pdf]])
* FF Clock Skew Timing Constraints ([[Media:CArch.ClockSkew.20131121.pdf |pdf]])

* Synchronizer ([[Media:CArch.Synchronizer.20131216.pdf |pdf]])
* Resolution Time Analysis  ([[Media:CArch.Resolution.20131216.pdf |pdf]])
&lt;/br&gt;

== '''Finite State Machine'''==

* FSM State Encoding 
* FSM Types : Mealy and Moore Machines
* FSM Example ([[Media:CArch.2.A.FSMExample.20141018.pdf |pdf]])
&lt;/br&gt;

== '''Array  Devices''' ==

=== ''' Memory Arrays '''=== 
* RAM 
** RAM Structure ([[Media:DD4.A.1.RAM.20131111.pdf|A.pdf]])
** RAM Timing ([[Media:DD4.B.1.RAMTiming.20131130.pdf|B.pdf]])
** FPGA RAM ([[Media:DD4.C.1.FPGARAM.20160513.pdf|C.pdf]])

* ROM
&lt;/br&gt;

=== ''' Logic Arrays '''=== 
* PLA 
* PAL
* PLD
* FPGA
** FPGA Structure
** FPGA Configuration ([[Media:DD4.C.1.FPGAConf.20131130.pdf|B.pdf]])
&lt;/br&gt;
&lt;/br&gt; 
[http://www.ece.cmu.edu/~ece548/localcpy/sramop.pdf  Synchronous SRAM Timing] &lt;/br&gt;
[http://www.micron.com/~/media/Documents/Products/Technical%20Note/DRAM/tn4529.pdf Asynchronous SRAM Timing]&lt;/br&gt;
[http://www.ece.cmu.edu/~ece548/localcpy/dramop.pdf DRAM Timing] &lt;/br&gt;

[http://www.ece.unm.edu/~jimp/415/slides/fpga_arch1.pdf FPGA Architectures] &lt;/br&gt;
[http://www.engr.siu.edu/~haibo/ece428/notes/ece428_fpgaarch.pdf CPLD &amp; FPGA] &lt;/br&gt;
&lt;/br&gt;

== ''' RTL Design Techniques''' ==
&lt;/br&gt;

''' Design Methodology '''
&lt;/br&gt;


''' Synthesis '''
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

== '''Logic Families and IOs''' ==
* BJT Based 
:: DTL (Diode-Transistor Logic)
:: TTL (Transistor-Transistor Logic)
:: ECL (Emitter-Coupled Logic)
 
* MOS Based
:: CMOS (Complementary MOS)
:: Pseudo-nMOS 
:: Transmission Gate
:: BiCMOS (Bipolr + CMOS)
 
* Dynamic CMOS 
:: Domino
:: Clocked-CMOS (C&lt;sup&gt;2&lt;/sup&gt;MOS)
&lt;/br&gt; 

* Modern I/O Standards
:: TTL and LVTTL (Low Voltage TTL)
:: CMOS and LVCMOS (Low Voltage CMOS)
:: SSTL (Stub Series Terminated Logic)
:: HSTL (High Speed Tranceiver Logic)
:: LVDS (Low Voltage Differential Signaling) 

&lt;/br&gt;

* Wikipedia Pages for Logic Families ([[Media:Logic Families.wiki.20140812.pdf|A.pdf]])
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&lt;[[The necessities in Computer Design]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Architecture]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Organization]]&gt;  &lt;/br&gt;


&lt;/br&gt; &lt;/br&gt;

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

== '''Old''' ==
'''Until 2011.12''' 
'''Chapter 1. Binary Numbers'''
* 1.1 Binary Numbers([[Media:BinaryNumbers.1.A.pdf|pdf]])

''' Minterm, Maxterm, HW '''
* 1.1 Lecture01([[Media:DigitalDesign.20110922.pdf|pdf]])

''' Overflow HW '''
* Overflow Table([[Media:Overflow table.20110924.pdf|pdf]])

''' K-Map '''
* K-Map([[Media:DigitalDesign.20110926.pdf|pdf]])

''' Binary Adder '''
* Binary Adder (C, S) ([[Media:DigitalDesign.20110929.pdf|pdf]])
* Overflow detection circuit (V) ([[Media:HW Overflow20111001.pdf|pdf]])

''' BCD to Ex3 Code Coversion, Dont' Care '''
* BCD to Ex3 Code Conversion ([[Media:DigitalDesign.20111006.pdf|pdf]])

''' Prime Implicant, Dont' Care '''
* Prime Implicant, Don't Care ([[Media:DigitalDesign.20111010.pdf|pdf]])
* HW 3.6 - explain the method of combining 0's and X's

''' Multiplexer / Demultiplexer '''
* Multiplexer ([[Media:DigitalDesign.20111024.pdf|pdf]])
* HW (TBD)

''' Flip Flop / Latch '''
* FF &amp; Latch ([[Media:DigitalDesign.20111027.pdf|pdf]])
* FF &amp; Latch HW ([[Media:DigitalDesign (HW).20111027.pdf|pdf]])
* Gated D Latch &amp; Master-Slave D FlipFlop ([[Media:DigitalDesign.20111031.pdf|pdf]])
* HW (Forbidden state and Indeterminate state) ([[Media:DigitalDesign (HW).20111102.pdf|pdf]]) (note in #2, S' R' instead of S R)
* Classical Edge Triggered D FlipFlop ([[Media:DigitalDesign.20111112.pdf|pdf]])
* HW (addition in SW and HW) ([[Media:DigitalDesign (HW).20111112.pdf|pdf]]) 
* FSM1  ([[Media:DigitalDesign.FSM1.20111117.pdf|pdf]]) 
* FSM2  ([[Media:DigitalDesign.FSM2.20111117.pdf|pdf]]) 
* HW (FSM Waveforms) ([[Media:DigitalDesign (HW).20111118.pdf|pdf]]) 
 
 
''' Counter '''
* Sychronous Counter ([[Media:DigitalDesign.20111121.pdf|pdf]])
* Ripple Counter, Multiplexer, Tri-state buffer([[Media:DigitalDesign.20111124.pdf|pdf]])
* Register ([[Media:DigitalDesign.register.20111201.pdf|pdf]])
* Timing ([[Media:DigitalDesign.timing.20111201.pdf|pdf]])
* HW (Multiplexer, Shift Register) ([[Media:DigitalDesign (HW).20111201.pdf|pdf]])
* Universal Shift Register, Memory Cell ([[Media:DigitalDesign.20111206.pdf|pdf]])
* HW (Bit Serial Adder) ([[Media:DigitalDesign (HW).20111206.pdf|pdf]])
 
 
''' Memory '''
* Memory ([[Media:DigitalDesign.20111208.pdf|pdf]])

''' Comparator, Multiplier '''
* Comparator, Multiplier ([[Media:DigitalDesign.20111219.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111219.draw.pdf|2.pdf]])
 
 
'''Multiplexer based design method '''
* Multiplexer Design Method ([[Media:DigitalDesign.20111221.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111221.draw.pdf|2.pdf]])


midterm result ([[Media:MidReult.20111027.pdf|pdf]])

 
* Edge Triggered Flip Flop ([[Media:EdgeTrigFF.20111224.pdf|pdf]])
* FF Timing ([[Media:FFTiming.20111203.pdf|pdf]])


&lt;/br&gt; &lt;/br&gt;
'''Until 2013.07''' 

''' Number Systems '''
* Binary Numbers ([[Media:DD.1.A.BinNum.20130309.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD.1.B.HexaNum.20130417.pdf|B.pdf]])
* Numbers in C programs ([[Media:DD.1.C.CNum.20130309.pdf|C.pdf]])
* Codes ([[Media:DD.1.D.Coding.20130319.pdf|pdf]])
&lt;/br&gt;

&lt;/br&gt;
* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|pdf]])
&lt;/br&gt;

''' Combinational Circuits '''
* Truth Tables and Boolean Functions ([[Media:DD.2.A.TTable.20130325.pdf|2A.pdf]])&lt;/br&gt;
* K-Map ([[Media:DD.2.A.KMap.20130329.pdf|2B.pdf]])&lt;/br&gt;
* Binary Addition in C ([[Media:DD.2.C.BAinC.20130329.pdf|2.C.pdf]])&lt;/br&gt;
* Binary Arithmetic ([[Media:DD.2.D.BAri.2013.pdf|2.D.pdf]])&lt;/br&gt;
* Boolean Algebra ([[Media:DD.2.E.BAlgebra.20130419.pdf|2.E.pdf]])&lt;/br&gt;
&lt;/br&gt;

''' Sequential Circuits '''
* Latches and Flip-flops ([[Media:DD.3.A.LatchFF.20130413.pdf|3A.pdf]])&lt;/br&gt;
* FSM (Finite State Machine) ([[Media:DD.3.B.FSM.20130417.pdf|3B.pdf]])&lt;/br&gt;
* SR Latch Forbidden State ([[Media:DD.3.C.FState.20130413.pdf|3C.pdf]])&lt;/br&gt;
* Flip-flop Timing ([[Media:DD.3.D.Timing.20130413.pdf|3D.pdf]])&lt;/br&gt;
* Metastability ([[Media:DD.3.E.MetaState.20130628.pdf|3E.pdf]])&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&quot;[[The necessities in Computer Design]]&quot;  &lt;/br&gt;
&quot;[[The necessities in Computer Architecture]]&quot;  &lt;/br&gt;

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>7ponhn49dnu90bspopx8uyhowm544fm</sha1>
    </revision>
    <revision>
      <id>2807828</id>
      <parentid>2807826</parentid>
      <timestamp>2026-05-06T17:39:14Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/*  Timing Analysis  */</comment>
      <origin>2807828</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9835" sha1="syyyj3i43ev8pp9m4aowp7m8x9ooy05" xml:space="preserve">
== ''' Number Systems '''== 

=== ''' Binary Representation '''=== 
* Binary Numbers ([[Media:DD1.1.A.BinaryNum.20130918.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD1.2.A.HexaNum.20130918.pdf|A.pdf]])
* Other Codes ([[Media:DD1.3A.Code.20250329.pdf|A.pdf]])

=== ''' Binary Arithmetic '''=== 
* Binary Arithmetic ([[Media:DD1.4.A.BinaryArith.20150425.pdf|A.pdf]])
* BCD Arithmetic ([[Media:DD1.5.A.BCDArith.20130918.pdf|A.pdf]])

=== ''' C Program Examples '''=== 
* Binary Numbers in C programs ([[Media:DD1.6.A.BNumInC.20140103.pdf|A.pdf]])
* Binary Addition in C programs ([[Media:DD1.7.A.BArithInC.20140103.pdf|A.pdf]])
&lt;/br&gt;

* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|C.pdf]])
&lt;/br&gt;

=== ''' Floating Point Numbers '''=== 

* Floating Point Representations  ([[Media:CDesign.5.A.FPoint.20140121.pdf|5A.pdf]])&lt;/br&gt;
:: See [http://www.iro.umontreal.ca/~aboulham/F1214/Session%206Arithm/Floating_Point_Numbers.pdf Floating Point Overview]
:: See [http://www.cs.auckland.ac.nz/~patrice/210-2006/210%20LN04_2.pdf Offset Binary Overview]
:: See [http://www.intersil.com/content/dam/Intersil/documents/an96/an9657.pdf Offset Binary &amp; Sin / Cosine]
:: See [http://www.ee.ic.ac.uk/hp/staff/dmb/courses/dig2/4_Analog.pdf Offset Binary &amp; ADC / DAC]
&lt;/br&gt;

=== ''' Interfacing Digital and Analog Signals '''=== 
* Sampling and Quantization   ([[Media:DD1.10.A.SampleQuant.20150425.pdf|A.pdf]])
* Digital-to-Analog Conversion  ([[Media:DD1.8.A.DAC.20140208.pdf|A.pdf]])
* Analog-to-Digital Conversion ([[Media:DD1.9.A.DAC.20140208.pdf|A.pdf]])
&lt;/br&gt;

== '''Combinational Circuits'''==

=== ''' Design '''=== 
* Boolean Algebra ([[Media:DD2.A1.BAlgebra.20250503.pdf|A1.pdf]])
* Truth Tables ([[Media:DD2.A2.TTable.20250424.pdf|A2.pdf]])
* K-Map ([[Media:DD2.A3.KMap.20250424.pdf|A3.pdf]])
* Design Examples ([[Media:DD2.A4.CombEx.20250414.pdf|A4.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Decoder ([[Media:DD2.B.1.Decoder.20130928.pdf|B1.pdf]])
* Encoder ([[Media:DD2.B.2.Encoder.20130917.pdf|B2.pdf]])
* Multiplexer ([[Media:DD2.B.3.Multiplexer.20130928.pdf|B3.pdf]])
* Adder ([[Media:DD2.B.4..Adder.20131007.pdf|B4.pdf]], [[Media:Fa.sch.20131002.pdf|fa.sch.pdf]], [[Media:Adder4.sch.20131002.pdf|adder4.sch.pdf]])
&lt;/br&gt;

=== ''' Design Metric '''=== 
* Noise Margin ([[Media:DD2.C1.NoiseMargin.20250415.pdf|C1.pdf]])
&lt;/br&gt;

== '''Sequential Circuits'''==

=== ''' Design '''=== 
* Types of Flip-Flops ([[Media:CDesign.1.A.FF.20130412.pdf |1A.pdf]])&lt;/br&gt;
* Latches and Flipflops ([[Media:DD3.A.1.LatchFF.20160308.pdf|A1.pdf]])
* State Transition Table ([[Media:DD3.A.2.pdf|A2.pdf]])
* FSM (Finite State Machine) ([[Media:DD3.A.3.FSM.20131030.pdf|A3.pdf]])
&lt;/br&gt;

* The Classic FF Design ([[Media:DD3.A.6.ClassicFF.20131126.pdf|A7.pdf]])
* The Modern FF Design ([[Media:DD3.A.6.ClassicFF.20131204.2.pdf|A8.pdf]])
&lt;/br&gt;

=== ''' Components '''=== 
* Latches and Flip-flops ([[Media:DD3.B.1.LatchFF.20131008.pdf|B1.pdf]])
* Registers ([[Media:DD3.B.2.Register.20150326.pdf|B2.pdf]], [[Media:Register.20131118.pdf|register.pdf]])
* Counters ([[Media:DD3.B.2.Counter.20150420.pdf|B3.pdf]])
&lt;/br&gt;

=== ''' Timing Analysis '''=== 
* Metastability ([[Media:DD3.A.4.MetaState.20131030.pdf|A4.pdf]])
* Flip-flop Timing ([[Media:DD3.A5.FFTiming.20260506.pdf|A5.pdf]])
* SR Latch Forbidden State ([[Media:DD3.A.5.ForbiddenState.20131030.pdf|A6.pdf]])
&lt;/br&gt;

* FF Min Max Timing Constraints ([[Media:CArch.MinMaxTiming.20131121.pdf |pdf]])
* FF Clock Skew Timing Constraints ([[Media:CArch.ClockSkew.20131121.pdf |pdf]])

* Synchronizer ([[Media:CArch.Synchronizer.20131216.pdf |pdf]])
* Resolution Time Analysis  ([[Media:CArch.Resolution.20131216.pdf |pdf]])
&lt;/br&gt;

== '''Finite State Machine'''==

* FSM State Encoding 
* FSM Types : Mealy and Moore Machines
* FSM Example ([[Media:CArch.2.A.FSMExample.20141018.pdf |pdf]])
&lt;/br&gt;

== '''Array  Devices''' ==

=== ''' Memory Arrays '''=== 
* RAM 
** RAM Structure ([[Media:DD4.A.1.RAM.20131111.pdf|A.pdf]])
** RAM Timing ([[Media:DD4.B.1.RAMTiming.20131130.pdf|B.pdf]])
** FPGA RAM ([[Media:DD4.C.1.FPGARAM.20160513.pdf|C.pdf]])

* ROM
&lt;/br&gt;

=== ''' Logic Arrays '''=== 
* PLA 
* PAL
* PLD
* FPGA
** FPGA Structure
** FPGA Configuration ([[Media:DD4.C.1.FPGAConf.20131130.pdf|B.pdf]])
&lt;/br&gt;
&lt;/br&gt; 
[http://www.ece.cmu.edu/~ece548/localcpy/sramop.pdf  Synchronous SRAM Timing] &lt;/br&gt;
[http://www.micron.com/~/media/Documents/Products/Technical%20Note/DRAM/tn4529.pdf Asynchronous SRAM Timing]&lt;/br&gt;
[http://www.ece.cmu.edu/~ece548/localcpy/dramop.pdf DRAM Timing] &lt;/br&gt;

[http://www.ece.unm.edu/~jimp/415/slides/fpga_arch1.pdf FPGA Architectures] &lt;/br&gt;
[http://www.engr.siu.edu/~haibo/ece428/notes/ece428_fpgaarch.pdf CPLD &amp; FPGA] &lt;/br&gt;
&lt;/br&gt;

== ''' RTL Design Techniques''' ==
&lt;/br&gt;

''' Design Methodology '''
&lt;/br&gt;


''' Synthesis '''
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

== '''Logic Families and IOs''' ==
* BJT Based 
:: DTL (Diode-Transistor Logic)
:: TTL (Transistor-Transistor Logic)
:: ECL (Emitter-Coupled Logic)
 
* MOS Based
:: CMOS (Complementary MOS)
:: Pseudo-nMOS 
:: Transmission Gate
:: BiCMOS (Bipolr + CMOS)
 
* Dynamic CMOS 
:: Domino
:: Clocked-CMOS (C&lt;sup&gt;2&lt;/sup&gt;MOS)
&lt;/br&gt; 

* Modern I/O Standards
:: TTL and LVTTL (Low Voltage TTL)
:: CMOS and LVCMOS (Low Voltage CMOS)
:: SSTL (Stub Series Terminated Logic)
:: HSTL (High Speed Tranceiver Logic)
:: LVDS (Low Voltage Differential Signaling) 

&lt;/br&gt;

* Wikipedia Pages for Logic Families ([[Media:Logic Families.wiki.20140812.pdf|A.pdf]])
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&lt;[[The necessities in Computer Design]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Architecture]]&gt;  &lt;/br&gt;
&lt;[[The necessities in Computer Organization]]&gt;  &lt;/br&gt;


&lt;/br&gt; &lt;/br&gt;

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

== '''Old''' ==
'''Until 2011.12''' 
'''Chapter 1. Binary Numbers'''
* 1.1 Binary Numbers([[Media:BinaryNumbers.1.A.pdf|pdf]])

''' Minterm, Maxterm, HW '''
* 1.1 Lecture01([[Media:DigitalDesign.20110922.pdf|pdf]])

''' Overflow HW '''
* Overflow Table([[Media:Overflow table.20110924.pdf|pdf]])

''' K-Map '''
* K-Map([[Media:DigitalDesign.20110926.pdf|pdf]])

''' Binary Adder '''
* Binary Adder (C, S) ([[Media:DigitalDesign.20110929.pdf|pdf]])
* Overflow detection circuit (V) ([[Media:HW Overflow20111001.pdf|pdf]])

''' BCD to Ex3 Code Coversion, Dont' Care '''
* BCD to Ex3 Code Conversion ([[Media:DigitalDesign.20111006.pdf|pdf]])

''' Prime Implicant, Dont' Care '''
* Prime Implicant, Don't Care ([[Media:DigitalDesign.20111010.pdf|pdf]])
* HW 3.6 - explain the method of combining 0's and X's

''' Multiplexer / Demultiplexer '''
* Multiplexer ([[Media:DigitalDesign.20111024.pdf|pdf]])
* HW (TBD)

''' Flip Flop / Latch '''
* FF &amp; Latch ([[Media:DigitalDesign.20111027.pdf|pdf]])
* FF &amp; Latch HW ([[Media:DigitalDesign (HW).20111027.pdf|pdf]])
* Gated D Latch &amp; Master-Slave D FlipFlop ([[Media:DigitalDesign.20111031.pdf|pdf]])
* HW (Forbidden state and Indeterminate state) ([[Media:DigitalDesign (HW).20111102.pdf|pdf]]) (note in #2, S' R' instead of S R)
* Classical Edge Triggered D FlipFlop ([[Media:DigitalDesign.20111112.pdf|pdf]])
* HW (addition in SW and HW) ([[Media:DigitalDesign (HW).20111112.pdf|pdf]]) 
* FSM1  ([[Media:DigitalDesign.FSM1.20111117.pdf|pdf]]) 
* FSM2  ([[Media:DigitalDesign.FSM2.20111117.pdf|pdf]]) 
* HW (FSM Waveforms) ([[Media:DigitalDesign (HW).20111118.pdf|pdf]]) 
 
 
''' Counter '''
* Sychronous Counter ([[Media:DigitalDesign.20111121.pdf|pdf]])
* Ripple Counter, Multiplexer, Tri-state buffer([[Media:DigitalDesign.20111124.pdf|pdf]])
* Register ([[Media:DigitalDesign.register.20111201.pdf|pdf]])
* Timing ([[Media:DigitalDesign.timing.20111201.pdf|pdf]])
* HW (Multiplexer, Shift Register) ([[Media:DigitalDesign (HW).20111201.pdf|pdf]])
* Universal Shift Register, Memory Cell ([[Media:DigitalDesign.20111206.pdf|pdf]])
* HW (Bit Serial Adder) ([[Media:DigitalDesign (HW).20111206.pdf|pdf]])
 
 
''' Memory '''
* Memory ([[Media:DigitalDesign.20111208.pdf|pdf]])

''' Comparator, Multiplier '''
* Comparator, Multiplier ([[Media:DigitalDesign.20111219.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111219.draw.pdf|2.pdf]])
 
 
'''Multiplexer based design method '''
* Multiplexer Design Method ([[Media:DigitalDesign.20111221.spread.pdf|1.pdf]], [[Media:DigitalDesign.20111221.draw.pdf|2.pdf]])


midterm result ([[Media:MidReult.20111027.pdf|pdf]])

 
* Edge Triggered Flip Flop ([[Media:EdgeTrigFF.20111224.pdf|pdf]])
* FF Timing ([[Media:FFTiming.20111203.pdf|pdf]])


&lt;/br&gt; &lt;/br&gt;
'''Until 2013.07''' 

''' Number Systems '''
* Binary Numbers ([[Media:DD.1.A.BinNum.20130309.pdf|A.pdf]])
* Hexadecimal Numbers ([[Media:DD.1.B.HexaNum.20130417.pdf|B.pdf]])
* Numbers in C programs ([[Media:DD.1.C.CNum.20130309.pdf|C.pdf]])
* Codes ([[Media:DD.1.D.Coding.20130319.pdf|pdf]])
&lt;/br&gt;

&lt;/br&gt;
* Helpful Wikipedia Pages ([[Media:DD.WP.NumberSystem.20130309.pdf|pdf]])
&lt;/br&gt;

''' Combinational Circuits '''
* Truth Tables and Boolean Functions ([[Media:DD.2.A.TTable.20130325.pdf|2A.pdf]])&lt;/br&gt;
* K-Map ([[Media:DD.2.A.KMap.20130329.pdf|2B.pdf]])&lt;/br&gt;
* Binary Addition in C ([[Media:DD.2.C.BAinC.20130329.pdf|2.C.pdf]])&lt;/br&gt;
* Binary Arithmetic ([[Media:DD.2.D.BAri.2013.pdf|2.D.pdf]])&lt;/br&gt;
* Boolean Algebra ([[Media:DD.2.E.BAlgebra.20130419.pdf|2.E.pdf]])&lt;/br&gt;
&lt;/br&gt;

''' Sequential Circuits '''
* Latches and Flip-flops ([[Media:DD.3.A.LatchFF.20130413.pdf|3A.pdf]])&lt;/br&gt;
* FSM (Finite State Machine) ([[Media:DD.3.B.FSM.20130417.pdf|3B.pdf]])&lt;/br&gt;
* SR Latch Forbidden State ([[Media:DD.3.C.FState.20130413.pdf|3C.pdf]])&lt;/br&gt;
* Flip-flop Timing ([[Media:DD.3.D.Timing.20130413.pdf|3D.pdf]])&lt;/br&gt;
* Metastability ([[Media:DD.3.E.MetaState.20130628.pdf|3E.pdf]])&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;
&lt;/br&gt;

See also &lt;/br&gt;
&quot;[[The necessities in Computer Design]]&quot;  &lt;/br&gt;
&quot;[[The necessities in Computer Architecture]]&quot;  &lt;/br&gt;

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>syyyj3i43ev8pp9m4aowp7m8x9ooy05</sha1>
    </revision>
  </page>
  <page>
    <title>Katie Harwood</title>
    <ns>0</ns>
    <id>124151</id>
    <revision>
      <id>2807845</id>
      <parentid>2786519</parentid>
      <timestamp>2026-05-06T20:13:44Z</timestamp>
      <contributor>
        <username>Tola73</username>
        <id>3020076</id>
      </contributor>
      <comment>/* Significance */ The page has had no significant updates since its creation in 2011. The inertia of time has allowed for greater perspective and new relevant info.</comment>
      <origin>2807845</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="27914" sha1="npo8l4w3ozlmfln2fnel8ckqzh1a0ad" xml:space="preserve">'''Katherine &quot;Katie&quot; Harwood''' is a fictional character in the 2002 film ''Ghost Ship'' where an innocent young girl goes on a sea voyage of a lifetime, only to be caught up in a living nightmare aboard the ill-fated ocean liner. In the film, Katie is the supporting deuteragonist to the main character (Maureen Epps) and stands in stark contrast to the completely evil and demonic antagonist (Jack Ferriman). In many regards, Katie is just as much of a heroine as Maureen Epps for enduring unfathomable suffering and risking the wrath of Jack Ferriman through her unyielding efforts to save the souls and lives of others on the ship.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; Katie is portrayed by a young [[w:Emily Browning|Emily Jane Browning]].

== Significance ==
Katie is an iconic representation of childhood from an earlier era, at a time when life was simpler, and childhood more innocent.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; &lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt; She stands out in a coarse mixed reviewed horror film as a compelling, distinctive and enduring character who subverts common horror child tropes.&lt;ref&gt;https://michellepatterson.net/2018/03/23/ghostship-film-review/Michelle Patterson Publications&lt;/ref&gt;&lt;ref&gt;https://moviefilmreview.com/186187/movie-review-of-ghost-ship-2002?utm_source=chatgpt.com/ Cal Knox Movie Review&lt;/ref&gt;&lt;ref&gt;https://lmariewood.com/2025/05/23/horror-tropes-when-to-use-them-and-when-to-subvert-them/ L. Marie Wood; Horror Tropes: When to Use Them and When to Subvert Them&lt;/ref&gt; With the Shout! Factory 20-year anniversary Blue-ray rerelease of ''Ghost Ship,'' Katie is made the new face of the film in the promotional artwork, elevating a morally intact child to iconic status; something very few horror children achieve, and even more so as a positive icon without villainy.&lt;ref&gt;https://www.blu-ray.com/news/?id=27285/ Shout! Factory Ghost Ship 20-year anniversary collector’s edition Blu-ray disc&lt;/ref&gt;

== Synopsis ==
In May of 1962, an endearing Katie waves goodbye to her grandparents in [[Europe]] and journeys solo aboard the exquisite ''Antonia Graza'' on an exclusive cruise to rejoin her family in New York City. Since Katie is the only child onboard, she receives special care and attention. In describing her voyage Katie states, ”The whole ship was my playground. I was the only child onboard, but the ship’s purser and captain took special care of me. I felt so safe and happy with them.”&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

However, after the Antonia Graza rescues a lone survivor and cargo from a sinking ship, Katie’s exciting journey suddenly takes a horrendous turn for the worse. Unbeknownst to Katie and her shipmates, the single survivor (Jack Ferriman) is literally a demonic henchman for Satan set on destroying lives and collecting souls. Ferriman informs select members of the crew about the millions in gold recovered from his sinking ship, and influences them to launch an elaborate plot to seize control of the Graza and the gold, by killing everyone onboard. Although several crewmembers and passengers try to save her life from the murderous conspirators, Katie is eventually caught and tragically hanged to death, with her body concealed behind the partitioning door of her cabin.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

Instead of ascending to her rightful place in heaven, Katie’s spirit is trapped on the ship by demonic forces along with the entire compliment of murdered passengers and crew. However, since Katie is still a young child, her soul is completely innocent and therefore beyond the influence and control of the demonic Ferriman. Over the next 40 years, the evil Ferriman along with the help of the wicked “marked” souls, try to use the ship as a trap to destroy unsuspecting lives, and collect a quota of souls for hell. Despite being the only flicker of good on the ship, Katie bravely opposes Ferriman and the evil spirits, and attempts to warn and save the lives of anyone who has the misfortune of being lured aboard the condemned vessel. Concerning these things, Katie says, ““Without the mark, Jack can’t control me and because of this, he hates me most of all; scaring me at every turn and chasing me away when I try to warn those who come here.”&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

When Maureen Epps and her ship salving crew are lured aboard the Graza by Ferriman, Katie attempts to warn them about the dangerous ship. Katie has very perceptive eyes, and finds Epps to be more open to the truth than her cohorts. Katie tries to leave Epps messages and clues, and even appears to her on several occasions. When Katie senses that Ferriman has sabotaged the salvage crew’s tugboat, she boldly tries to warn the crew, but is forcibly carried away by Ferriman. Epps witnesses Katie’s warning and sets off in search for the mysterious little girl.

Epps locates Katie’s cabin and comes face to face with her remains and her spirit. Katie allows Epps to have her cherished heart-shaped locket, and then proceeds to speak directly about the ship being a demonic trap. However, before Katie can finish explaining, she is overheard by an invisible Ferriman and lets out a frightened scream as he removes her from the cabin. Undeterred, Katie returns and tries to help Epps and her remaining crewmembers escape. Endowed with supernatural power from above, Katie transports Epps back to May 21, 1962 through her memories and reveals to her the horrifying events that led to everyone’s death, including her own. Furthermore, Katie reveals the true identity of Jack Ferriman and his goal to use the ship as a conduit to collect souls for Satan.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

After experiencing Katie’s heartbreaking vision, Epps decides to risk almost certain death by destroying the cursed ship with explosives. However, Katie helps Epps escape the rapidly sinking vessel, while emancipated souls rise from the ship. Katie stays faithfully by Epps and gives her a grateful smile as her spirit ascends toward heaven.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

== Religious &amp; Symbolic Implications ==
At its core, the story is deeply religious and the plot hinges on the concept of a young girl trying to save the lives (and ultimately souls) of others from demonic forces.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Metaphorical Murals=====
The religious implications and [[Christian symbolism]] surrounding Katie’s story are rich. &lt;ref&gt;http://www.worsleyschool.net/socialarts/symbolism/page.html&lt;/ref&gt; Although only lightly touched on in the final version of the film, the most obvious religious connections come through the [[Gustave Dore]] inspired murals displayed throughout the ship. The murals are depictions based on Dante Alighieri’s Divine Comedy and The Inferno. The metaphorical murals hint at the struggle between Katie and Ferriman over the lives and souls on the ship.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; Lead actress Julianna Margulies says, “Katie is like Virgil in Dante’s Inferno. Margulies explains that Katie is like a “little guide” trying to lead others through hell and then safely onto the other side.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; As in the mural of Charon, Ferriman is literally a “ferry man” using a boat in the attempt to ferry souls to hell and eternal damnation.  Conversely, Katie opposes Ferriman by trying to protect and warn all unsuspecting visitors and sharing the truth about the demonic ship.

=====Marked Souls=====
An additional concept presented in the plot are the “marked” souls. Like the murals in the story, the souls of the sinful and the “lost”&lt;ref&gt;{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN |page=563 | year=1983}}&lt;/ref&gt; are marked with a hooked-shaped indention on their hand. Katie says the mark is “a sign of their sins” and that they are bound to the will of Satan and his demons, in addition to eventually sharing in their same flaming judgment.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; However, not all the murals are somber. The depiction of angels defeating the demons and casting them down foreshadows the eventual defeat of Ferriman and ultimately all evil.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====The Dove and Locket=====
Another subtle, but implicit [[Christian symbolism]] is Katie’s heart-shaped locket with the image of a raised dove in flight.&lt;ref&gt;{{cite book| last=Barton |first=John | title=The Oxford Bible Commentary |publisher=Oxford University Press |location=New York, NY  | year=2001 |page=383}}&lt;/ref&gt; In Christian tradition, this type of dove can symbolize innocence, purity, faith, hope, God’s peace, God’s presence through the [[Holy Spirit (Christianity)|Holy Spirit]] and God’s guidance and deliverance.&lt;ref&gt;http://www.newadvent.org/cathen/05144b.htm&lt;/ref&gt; The dove also indicated God’s bestowal of fortitude necessary to bear suffering and death&lt;ref&gt;{{cite book| last=Gauding |first=Madonna | title=The Signs and Symbols Bible |publisher=Sterling Publishing Co |location=New York, NY  | year=2009 |page=82}}&lt;/ref&gt;. In this sense, the remains of the dead dove on the bridge of the ship, foreshadows Katie’s death, but at the same time indicates that there could still be hope.  When Epps puts on Katie’s locket, it implies that Epps now has a renewed heart, or possibly even salvation as stated in Bible scripture. In this manner, the locket could also represent that Epps’ soul is not &quot;marked&quot;.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Parallel of an Innocent and Sacrificial Life=====
A striking parallel could also be drawn between Katie and Jesus Christ. In the bible, Christ lives a completely innocent life, but is hung on a cross to sacrificially die for the sins of all humanity.&lt;ref&gt;{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN  | year=1983 |pages=2228-2229}}&lt;/ref&gt; Christ tastes the sting of death and hell in order to offer salvation to all who will put their faith in Him.&lt;ref&gt;{{cite book| last=Barton |first=John | title=The Oxford Bible Commentary |publisher=Oxford University Press |location=New York, NY  | year=2001 |page=966}}&lt;/ref&gt; Similarly, innocent Katie is hung by a rope behind a dividing door, but through her death she is able to lead others to &quot;salvation&quot; and safety. Because of her young age and innocent death, Katie is the only voice of hope for all the suppressed souls held captive on the demonic ship. Katie also endures small tastes of hell from Ferriman in her efforts to save others. However, Katie’s love and compassion for the lives of others is greater than her fear and pain, and in similarity to Christ, Katie rises victoriously to heaven in the end.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Forty Years and the Promised Land=====
40 years is a significant reoccurring number in the Christian faith.&lt;ref&gt;Collins English Dictonary http://dictionary.reference.com/browse/promised land&lt;/ref&gt; The [[Israelites]] endured 40 years of great trial and testing in the wilderness before they were able to enter their promised land of rest. In Christian tradition, the earthly life is often compared to a time of trial in the wilderness and the Promised Land a metaphor for heaven.&lt;ref&gt;Collins English Dictonary http://dictionary.reference.com/browse/promised land&lt;/ref&gt; Correspondingly, Katie has endured the “wilderness” and trials of the ship for 40 years, but as the result of Epps’ help she is inevitably freed to enter the “promised land” of heaven.

=====Concept of Free Will=====
Free will is both discussed and demonstrated in the story.&lt;ref&gt;{{cite book| last=Keck |first=Leander | title=The New Interpreter's  Bible vol. 9 |publisher=Abingdon Press |location=Nashville, TN  | year=2002 |pages=61-62}}&lt;/ref&gt; Ultimately, the crew members who die are led to destruction by their own sins and their selfish desire for greed, lust and power. Epps is spared the fate of her shipmates through listening to Katie’s warnings and humbly responding to the truth.&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

=====Elevated Status of Children=====
In the Christian bible God often uses the most humble and unlikely people to work through &lt;ref&gt;{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN  | year=1983 |page=2283}}&lt;/ref&gt; – Out of hundreds on the ship, Katie is the meekest, humblest and most unlikely to do anything significant, yet every soul on the ship ends up depending on her.&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page production notes]&lt;/ref&gt;

Katie is also reminiscent of Christ elevating the status of children when he gave them special attention and profoundly declared that unless one humbles them self and has the faith of a little child, they will not enter the kingdom of heaven. This is further illustrated when Christ took a lowly child and placed him in the midst of his followers and stated that only those with childlike faith will be the greatest in the kingdom of heaven.&lt;ref&gt;{{cite book| last=Barton |first=John | title=The Oxford Bible Commentary |publisher=Oxford University Press |location=New York, NY  | year=2001 |page=963}}&lt;/ref&gt; This concept is demonstrated by Epps' willingness to humble herself to Katie's level.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

The role of children in Christian scripture – Comparisons could be made between Katie and several unpretentious bible characters such as the lowly shepherd boy David who defeated the warrior Goliath and became king. Outwardly both young Katie and David appear to be nothing more that mere children, however, inwardly they possess vast unforeseen potential and a tremendous strength of character.&lt;ref&gt;{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN  | year=1983 |page=563}}&lt;/ref&gt; &lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Eternal Destinations=====
In the film, as with [[Christian soteriology]] there are two clear eternal destinations which souls go to after death; ether heaven or hell.  A heart of child-like faith will put one on the path to heaven, but a sinful life eventually leads to destruction.&lt;ref&gt;{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN  | year=1983 |page=1902}}&lt;/ref&gt;

=====The Destructiveness of Sinful Choices=====
One of the morals of the story is the corrupting dangers of pursuing riches. This theme also fits aptly within the religious concepts of the film.&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt; Sinful greed, lust, debauchery and selfish desire can influence ordinarily good people to do horribly evil things (1Timothy 6:9-10).&lt;ref&gt;{{cite book| title=Life Application Bible - New International Version |publisher=Tyndale House Publishers, Inc. |location=Wheaton, IL | page=2195 | year=1991}}&lt;/ref&gt; The choices made by the characters in the story illustrate this principle.

“Be sure your sins will find you out” – As the story goes, sinful activities have consequences and will eventually have to be accounted for, and the ultimate payment for sin in the end is both physical and spiritual death (Romans 6:20-23). &lt;ref&gt;{{cite book| title=Life Application Bible - New International Version |publisher=Tyndale House Publishers, Inc. |location=Wheaton, IL | page=2039 | year=1991}}&lt;/ref&gt; &lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

=====Loving Others before Ourselves=====
“Greater love has no one than this – To lay one’s life down for their friends.” Katie and Epps are both willing put themselves at risk to save lives. Through the process of trying to help others both Epps and Katie experience help for themselves.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

“Those who seek to save their life will lose it, but those who are willing to lose their life will save it.”&lt;ref&gt;{{cite book| title=Life Application Bible - New International Version |publisher=Tyndale House Publishers, Inc. |location=Wheaton, IL | page=1842 | year=1991}}&lt;/ref&gt; In this sense Epps' life is saved, because she places the needs of others first. Katie also inherits eternal life in heaven by risking herself to help others.

=====Purgatory=====
The concept of purgatory (Whalen p. 1034-1039)&lt;ref&gt;{{cite book| last=Whalen |first=John | title=New Catholic Encyclopedia Vol. 11|publisher=Catholic University of America |location=Washington, D.C.  | year=1983}}&lt;/ref&gt; is also mentioned when Katie states that the ship had become like a “prison” where souls are trapped among the living between heaven and hell.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

====Additional Symbolism====
=====Life as a Voyage on a Ship=====
Life itself is comparable to a voyage on ship...illustrating that our lives are like a ship that is bound for one of two possible destinations.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; In the context of Ghost Ship, the voyage ends in destruction (death and hell) unless we get off. In this sense, Katie pleads with us to get off, but that involves humbly changing our ways and giving up the allure and pursuit of vain things such as wealth and power.&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

=====Symbols of Changing Times=====
The once elegant and decaying ship is a symbol of changing times.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; In a similar manner, Katie can be viewed as a symbolic representation of childhood from a more innocent era. Just as the ship decays to ruin from its glorious 1950’s splendor, Katie’s deteriorating childhood possessions and even Katie herself, are evocative symbols of changing times and eroding childhood innocence.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

== Dichotomy and Contrasts ==
Katie is in some ways a complicated paradox: She has been deeply hurt and traumatized by all she has experienced, but by the same token, she is still very much an innocent child – Even after 40 years of resilient growth as a character. Katie exhibits a wide array of emotions from timid youthful innocence and joy, to extreme terror, sadness, loneliness, and solemn distress.&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

====Characters====
Even under tremendously trying circumstances Katie demonstrates an incredible inner strength of character without compromising the sweet kind-hearted &quot;girly&quot; girl she really is. The fact that Katie remains true to herself in spite of all the evil and suffering she experienced transcends their eroding influence and distinguishes her from the other characters.&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

=====Ferriman=====
Ferriman is representative of Satan and the crafty deceptiveness of evil in destroying people’s lives. Katie and Ferriman are at completely opposite ends of the Christian spectrum – The demonic Ferriman is totally depraved and evil while Katie is sweet and innocent. Katie is motivated by love and compassion, Ferriman is motivated by hate and rage.&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt; &lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Epps=====
Although Katie and Epps are two completely dissimilar characters from different eras, they still grow quite close. Epps is &quot;a woman working in a man's world&quot;&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt; and appears somewhat tough and masculine on the exterior like a tomboy. Although not specifically stated, the story indicates that Epps is likely from a broken home. On the other hand, Katie comes from a close, traditional nuclear family. In contrast to Epps, Katie is more of a typical &quot;girly&quot; girl and is slightly timid, but possess a stout inner fortitude despite her diminutive and delicate appearance.&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

=====Francesca=====
The sultry Francesca lives the more representative life of a performing artist. Unlike Epps who feels the need to downplay her femininity, Francesca leans toward the other extreme and attempts to flaunt her femininity to her own advantage, even to the point of her own exploitation. Conversely, Katie is still a virtuous and modest little girl, who has yet to be tainted by any corrupting influences of the world.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Captain Murphy=====
Captain Murphy appears to be a strong imposing ship captain on the outside, however, when confronted with trials and tribulations, his perceived strength is shown to be superficial. This is evident by his emotional breakdown where he shuts off emotionally and withdraws to himself while attempting to drown his problems in alcohol. In contrast, Katie appears to be very delicate in both physical appearance and personality. However, despite her dainty appearance, Katie proves her true strength and fortitude from within through dealing with her pain and problems directly, instead of allow it to define her&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

====General Themes====
Light vs. darkness, good vs. evil, childhood innocence vs. wickedness, tragedy vs. triumph, simpler more wholesome times, vs. the complexity and loss of innocence of today.&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

== Conclusion ==
On the surface, Katie’s story is an emotional tale of heartache and tragedy – Something that no young person should ever have to endure. However, Katie’s story is also essential to the plot. &lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt; Through helping each other and being willing to put themselves at risk, Katie and Epps together achieve a triumph out of the tragedy and deliver a major blow to Ferriman and the forces of evil.  Although the struggle over the souls of people will continue, this victory of the human spirit gives hope even in the face of terrible suffering, evil and even death.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

“Katie is really a very sweet girl.  She's completely innocent…“ says Emily Jane Browning, who obviously became completely immersed in her character.  “…She's been hoping someone would come onto the ship to be her friend, so when Epps arrives she's very excited – they develop a real friendship.&quot;&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

”Emily did an amazing job,” Director Steve Beck enthuses. “She gave Katie a real complexity…she's not just a little girl caught up in a ghost story…”  Both Browning and Beck also added that underneath all her heartfelt emotions, Katie longs for vindication and justice.&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

The deeply symbolic and metaphorical aspects surrounding Katie’s story along with the close relationships she develops with others lends support to the notion that the film was originally intended to be more than just another Hollywood “slasher” horror. This is further substantiated by Katie’s tearful and heartfelt story – Emotionally poignant and touching drama not usually associated with a horror picture.&lt;ref&gt;http://www.dailyscript.com/scripts/ghost_ship_info.txt&lt;/ref&gt;

== References ==
===Footnotes===
{{Reflist|colwidth=30em}}

===Cited Texts===
{{refbegin}}

*{{cite book| last=Barclay | first=William | title=The Gospel of Mark |publisher=Westminster Press | location=Philadelphia, PA | isbn=0664213022 | year=1975}}
*{{cite book| last=Barton |first=John | title=The Oxford Bible Commentary |publisher=Oxford University Press |location=New York, NY  | isbn=0-19-875500-7 | year=2001}}
*{{cite book| last=Boettner |first=Loraine | title=Roman Catholicism |publisher=Presbyterian &amp; Reformed Publishing Co. |location=Phillipsburg, NJ |isbn=0875521304 | year=1985}}
*{{cite book| last=Buttrick |first=George | title=The Interpreter's Bible vol. 12 |publisher=Abingdon Press |location=Nashville, TN |asin=B000HTP248 | year=1957}}
*{{cite book |title=Collins English Dictionary - Complete &amp; Unabridged 10th Edition  |date=2009 | publisher=Harper Collins |location=New York, NY}}
*{{cite book| last=Dummelow |first=J.R. | title=Commentary on the Whole Bible |publisher=Macmillian Publishing Co. |location=New York, NY | year=1936}}
*{{cite book| last=Gaebelein |first=Frank | title=The Expositors Bible Commentary vol. 1 |publisher=Zondervan |location=Grand Rapids, MI |isbn=0310364302 | year=1979}}
*{{cite book| last=Gauding |first=Madonna | title=The Signs and Symbols Bible |publisher=Sterling Publishing Co. |location=New York, NY |isbn=1402770049| year=2009}}
*{{cite book| last=Jamieson |first=Fausset | title=Commentary on the Whole Bible  |publisher=Zondervan |location=Grand Rapids, MI| asin=B004BCQP8O | year=1971}}
*{{cite book| last=Keck |first=Leander | title=New Interpreter's Bible vol. 9 |publisher=Abingdon Press |location=Nashville, TN |isbn=0687278228 | year=2002}}
*{{cite book| title=Life Application Bible - New International Version |publisher=Tyndale House Publishers, Inc. |location=Wheaton, IL | lccn=90-71553| year=1991}}
*{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN |isbn=0840752954 | year=1983}}
*{{cite book| last=Wall |first=Robert | title=The New Interpreter's Bible vol. 10 |publisher=Abingdon Press |location=Nashville, TN |isbn=0687278236 | year=2002}}
*{{cite book| last=Whalen |first=John | title=New Catholic Encyclopedia Vol. 11 |publisher=Catholic University of America |location=Washington D.C. | lccn=66-22292 | year=1967}}

{{refend}}

== External links ==
* [http://www.dailyscript.com/scripts/ghost_ship_info.txt/ Daily Script_Ghost Ship]
* [http://www.imdb.com/title/tt0288477/ Internet Movie Database_Ghost Ship 2002]
* [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]
* [http://www.rottentomatoes.com/m/ghost_ship/ Rotten Tomatoes movie review_Ghost Ship]
[[Category:Analysis]]
[[Category:Film]]</text>
      <sha1>npo8l4w3ozlmfln2fnel8ckqzh1a0ad</sha1>
    </revision>
    <revision>
      <id>2807846</id>
      <parentid>2807845</parentid>
      <timestamp>2026-05-06T20:29:17Z</timestamp>
      <contributor>
        <username>Tola73</username>
        <id>3020076</id>
      </contributor>
      <minor />
      <comment>/* Significance */ Minor wording change</comment>
      <origin>2807846</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="27928" sha1="np2a533cucx2oqglbfc1x4kn4uzknq8" xml:space="preserve">'''Katherine &quot;Katie&quot; Harwood''' is a fictional character in the 2002 film ''Ghost Ship'' where an innocent young girl goes on a sea voyage of a lifetime, only to be caught up in a living nightmare aboard the ill-fated ocean liner. In the film, Katie is the supporting deuteragonist to the main character (Maureen Epps) and stands in stark contrast to the completely evil and demonic antagonist (Jack Ferriman). In many regards, Katie is just as much of a heroine as Maureen Epps for enduring unfathomable suffering and risking the wrath of Jack Ferriman through her unyielding efforts to save the souls and lives of others on the ship.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; Katie is portrayed by a young [[w:Emily Browning|Emily Jane Browning]].

== Significance ==
Katie is an iconic representation of childhood from an earlier era, at a time when life was simpler, and childhood more innocent.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; &lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt; She stands out in a coarse mixed reviewed horror film as a compelling, distinctive and enduring character who subverts common horror child tropes.&lt;ref&gt;https://michellepatterson.net/2018/03/23/ghostship-film-review/Michelle Patterson Publications&lt;/ref&gt;&lt;ref&gt;https://moviefilmreview.com/186187/movie-review-of-ghost-ship-2002?utm_source=chatgpt.com/ Cal Knox Movie Review&lt;/ref&gt;&lt;ref&gt;https://lmariewood.com/2025/05/23/horror-tropes-when-to-use-them-and-when-to-subvert-them/ L. Marie Wood; Horror Tropes: When to Use Them and When to Subvert Them&lt;/ref&gt; With the Shout! Factory 20-year anniversary Blue-ray rerelease of ''Ghost Ship,'' Katie is made the new face of the film in the promotional artwork, elevating a morally intact child to iconic status; something very few horror children achieve, and even more so as a tragic and sympathetic icon without villainy.&lt;ref&gt;https://www.blu-ray.com/news/?id=27285/ Shout! Factory Ghost Ship 20-year anniversary collector’s edition Blu-ray disc&lt;/ref&gt;

== Synopsis ==
In May of 1962, an endearing Katie waves goodbye to her grandparents in [[Europe]] and journeys solo aboard the exquisite ''Antonia Graza'' on an exclusive cruise to rejoin her family in New York City. Since Katie is the only child onboard, she receives special care and attention. In describing her voyage Katie states, ”The whole ship was my playground. I was the only child onboard, but the ship’s purser and captain took special care of me. I felt so safe and happy with them.”&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

However, after the Antonia Graza rescues a lone survivor and cargo from a sinking ship, Katie’s exciting journey suddenly takes a horrendous turn for the worse. Unbeknownst to Katie and her shipmates, the single survivor (Jack Ferriman) is literally a demonic henchman for Satan set on destroying lives and collecting souls. Ferriman informs select members of the crew about the millions in gold recovered from his sinking ship, and influences them to launch an elaborate plot to seize control of the Graza and the gold, by killing everyone onboard. Although several crewmembers and passengers try to save her life from the murderous conspirators, Katie is eventually caught and tragically hanged to death, with her body concealed behind the partitioning door of her cabin.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

Instead of ascending to her rightful place in heaven, Katie’s spirit is trapped on the ship by demonic forces along with the entire compliment of murdered passengers and crew. However, since Katie is still a young child, her soul is completely innocent and therefore beyond the influence and control of the demonic Ferriman. Over the next 40 years, the evil Ferriman along with the help of the wicked “marked” souls, try to use the ship as a trap to destroy unsuspecting lives, and collect a quota of souls for hell. Despite being the only flicker of good on the ship, Katie bravely opposes Ferriman and the evil spirits, and attempts to warn and save the lives of anyone who has the misfortune of being lured aboard the condemned vessel. Concerning these things, Katie says, ““Without the mark, Jack can’t control me and because of this, he hates me most of all; scaring me at every turn and chasing me away when I try to warn those who come here.”&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

When Maureen Epps and her ship salving crew are lured aboard the Graza by Ferriman, Katie attempts to warn them about the dangerous ship. Katie has very perceptive eyes, and finds Epps to be more open to the truth than her cohorts. Katie tries to leave Epps messages and clues, and even appears to her on several occasions. When Katie senses that Ferriman has sabotaged the salvage crew’s tugboat, she boldly tries to warn the crew, but is forcibly carried away by Ferriman. Epps witnesses Katie’s warning and sets off in search for the mysterious little girl.

Epps locates Katie’s cabin and comes face to face with her remains and her spirit. Katie allows Epps to have her cherished heart-shaped locket, and then proceeds to speak directly about the ship being a demonic trap. However, before Katie can finish explaining, she is overheard by an invisible Ferriman and lets out a frightened scream as he removes her from the cabin. Undeterred, Katie returns and tries to help Epps and her remaining crewmembers escape. Endowed with supernatural power from above, Katie transports Epps back to May 21, 1962 through her memories and reveals to her the horrifying events that led to everyone’s death, including her own. Furthermore, Katie reveals the true identity of Jack Ferriman and his goal to use the ship as a conduit to collect souls for Satan.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

After experiencing Katie’s heartbreaking vision, Epps decides to risk almost certain death by destroying the cursed ship with explosives. However, Katie helps Epps escape the rapidly sinking vessel, while emancipated souls rise from the ship. Katie stays faithfully by Epps and gives her a grateful smile as her spirit ascends toward heaven.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

== Religious &amp; Symbolic Implications ==
At its core, the story is deeply religious and the plot hinges on the concept of a young girl trying to save the lives (and ultimately souls) of others from demonic forces.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Metaphorical Murals=====
The religious implications and [[Christian symbolism]] surrounding Katie’s story are rich. &lt;ref&gt;http://www.worsleyschool.net/socialarts/symbolism/page.html&lt;/ref&gt; Although only lightly touched on in the final version of the film, the most obvious religious connections come through the [[Gustave Dore]] inspired murals displayed throughout the ship. The murals are depictions based on Dante Alighieri’s Divine Comedy and The Inferno. The metaphorical murals hint at the struggle between Katie and Ferriman over the lives and souls on the ship.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; Lead actress Julianna Margulies says, “Katie is like Virgil in Dante’s Inferno. Margulies explains that Katie is like a “little guide” trying to lead others through hell and then safely onto the other side.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; As in the mural of Charon, Ferriman is literally a “ferry man” using a boat in the attempt to ferry souls to hell and eternal damnation.  Conversely, Katie opposes Ferriman by trying to protect and warn all unsuspecting visitors and sharing the truth about the demonic ship.

=====Marked Souls=====
An additional concept presented in the plot are the “marked” souls. Like the murals in the story, the souls of the sinful and the “lost”&lt;ref&gt;{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN |page=563 | year=1983}}&lt;/ref&gt; are marked with a hooked-shaped indention on their hand. Katie says the mark is “a sign of their sins” and that they are bound to the will of Satan and his demons, in addition to eventually sharing in their same flaming judgment.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; However, not all the murals are somber. The depiction of angels defeating the demons and casting them down foreshadows the eventual defeat of Ferriman and ultimately all evil.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====The Dove and Locket=====
Another subtle, but implicit [[Christian symbolism]] is Katie’s heart-shaped locket with the image of a raised dove in flight.&lt;ref&gt;{{cite book| last=Barton |first=John | title=The Oxford Bible Commentary |publisher=Oxford University Press |location=New York, NY  | year=2001 |page=383}}&lt;/ref&gt; In Christian tradition, this type of dove can symbolize innocence, purity, faith, hope, God’s peace, God’s presence through the [[Holy Spirit (Christianity)|Holy Spirit]] and God’s guidance and deliverance.&lt;ref&gt;http://www.newadvent.org/cathen/05144b.htm&lt;/ref&gt; The dove also indicated God’s bestowal of fortitude necessary to bear suffering and death&lt;ref&gt;{{cite book| last=Gauding |first=Madonna | title=The Signs and Symbols Bible |publisher=Sterling Publishing Co |location=New York, NY  | year=2009 |page=82}}&lt;/ref&gt;. In this sense, the remains of the dead dove on the bridge of the ship, foreshadows Katie’s death, but at the same time indicates that there could still be hope.  When Epps puts on Katie’s locket, it implies that Epps now has a renewed heart, or possibly even salvation as stated in Bible scripture. In this manner, the locket could also represent that Epps’ soul is not &quot;marked&quot;.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Parallel of an Innocent and Sacrificial Life=====
A striking parallel could also be drawn between Katie and Jesus Christ. In the bible, Christ lives a completely innocent life, but is hung on a cross to sacrificially die for the sins of all humanity.&lt;ref&gt;{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN  | year=1983 |pages=2228-2229}}&lt;/ref&gt; Christ tastes the sting of death and hell in order to offer salvation to all who will put their faith in Him.&lt;ref&gt;{{cite book| last=Barton |first=John | title=The Oxford Bible Commentary |publisher=Oxford University Press |location=New York, NY  | year=2001 |page=966}}&lt;/ref&gt; Similarly, innocent Katie is hung by a rope behind a dividing door, but through her death she is able to lead others to &quot;salvation&quot; and safety. Because of her young age and innocent death, Katie is the only voice of hope for all the suppressed souls held captive on the demonic ship. Katie also endures small tastes of hell from Ferriman in her efforts to save others. However, Katie’s love and compassion for the lives of others is greater than her fear and pain, and in similarity to Christ, Katie rises victoriously to heaven in the end.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Forty Years and the Promised Land=====
40 years is a significant reoccurring number in the Christian faith.&lt;ref&gt;Collins English Dictonary http://dictionary.reference.com/browse/promised land&lt;/ref&gt; The [[Israelites]] endured 40 years of great trial and testing in the wilderness before they were able to enter their promised land of rest. In Christian tradition, the earthly life is often compared to a time of trial in the wilderness and the Promised Land a metaphor for heaven.&lt;ref&gt;Collins English Dictonary http://dictionary.reference.com/browse/promised land&lt;/ref&gt; Correspondingly, Katie has endured the “wilderness” and trials of the ship for 40 years, but as the result of Epps’ help she is inevitably freed to enter the “promised land” of heaven.

=====Concept of Free Will=====
Free will is both discussed and demonstrated in the story.&lt;ref&gt;{{cite book| last=Keck |first=Leander | title=The New Interpreter's  Bible vol. 9 |publisher=Abingdon Press |location=Nashville, TN  | year=2002 |pages=61-62}}&lt;/ref&gt; Ultimately, the crew members who die are led to destruction by their own sins and their selfish desire for greed, lust and power. Epps is spared the fate of her shipmates through listening to Katie’s warnings and humbly responding to the truth.&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

=====Elevated Status of Children=====
In the Christian bible God often uses the most humble and unlikely people to work through &lt;ref&gt;{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN  | year=1983 |page=2283}}&lt;/ref&gt; – Out of hundreds on the ship, Katie is the meekest, humblest and most unlikely to do anything significant, yet every soul on the ship ends up depending on her.&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page production notes]&lt;/ref&gt;

Katie is also reminiscent of Christ elevating the status of children when he gave them special attention and profoundly declared that unless one humbles them self and has the faith of a little child, they will not enter the kingdom of heaven. This is further illustrated when Christ took a lowly child and placed him in the midst of his followers and stated that only those with childlike faith will be the greatest in the kingdom of heaven.&lt;ref&gt;{{cite book| last=Barton |first=John | title=The Oxford Bible Commentary |publisher=Oxford University Press |location=New York, NY  | year=2001 |page=963}}&lt;/ref&gt; This concept is demonstrated by Epps' willingness to humble herself to Katie's level.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

The role of children in Christian scripture – Comparisons could be made between Katie and several unpretentious bible characters such as the lowly shepherd boy David who defeated the warrior Goliath and became king. Outwardly both young Katie and David appear to be nothing more that mere children, however, inwardly they possess vast unforeseen potential and a tremendous strength of character.&lt;ref&gt;{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN  | year=1983 |page=563}}&lt;/ref&gt; &lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Eternal Destinations=====
In the film, as with [[Christian soteriology]] there are two clear eternal destinations which souls go to after death; ether heaven or hell.  A heart of child-like faith will put one on the path to heaven, but a sinful life eventually leads to destruction.&lt;ref&gt;{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN  | year=1983 |page=1902}}&lt;/ref&gt;

=====The Destructiveness of Sinful Choices=====
One of the morals of the story is the corrupting dangers of pursuing riches. This theme also fits aptly within the religious concepts of the film.&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt; Sinful greed, lust, debauchery and selfish desire can influence ordinarily good people to do horribly evil things (1Timothy 6:9-10).&lt;ref&gt;{{cite book| title=Life Application Bible - New International Version |publisher=Tyndale House Publishers, Inc. |location=Wheaton, IL | page=2195 | year=1991}}&lt;/ref&gt; The choices made by the characters in the story illustrate this principle.

“Be sure your sins will find you out” – As the story goes, sinful activities have consequences and will eventually have to be accounted for, and the ultimate payment for sin in the end is both physical and spiritual death (Romans 6:20-23). &lt;ref&gt;{{cite book| title=Life Application Bible - New International Version |publisher=Tyndale House Publishers, Inc. |location=Wheaton, IL | page=2039 | year=1991}}&lt;/ref&gt; &lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

=====Loving Others before Ourselves=====
“Greater love has no one than this – To lay one’s life down for their friends.” Katie and Epps are both willing put themselves at risk to save lives. Through the process of trying to help others both Epps and Katie experience help for themselves.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

“Those who seek to save their life will lose it, but those who are willing to lose their life will save it.”&lt;ref&gt;{{cite book| title=Life Application Bible - New International Version |publisher=Tyndale House Publishers, Inc. |location=Wheaton, IL | page=1842 | year=1991}}&lt;/ref&gt; In this sense Epps' life is saved, because she places the needs of others first. Katie also inherits eternal life in heaven by risking herself to help others.

=====Purgatory=====
The concept of purgatory (Whalen p. 1034-1039)&lt;ref&gt;{{cite book| last=Whalen |first=John | title=New Catholic Encyclopedia Vol. 11|publisher=Catholic University of America |location=Washington, D.C.  | year=1983}}&lt;/ref&gt; is also mentioned when Katie states that the ship had become like a “prison” where souls are trapped among the living between heaven and hell.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

====Additional Symbolism====
=====Life as a Voyage on a Ship=====
Life itself is comparable to a voyage on ship...illustrating that our lives are like a ship that is bound for one of two possible destinations.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; In the context of Ghost Ship, the voyage ends in destruction (death and hell) unless we get off. In this sense, Katie pleads with us to get off, but that involves humbly changing our ways and giving up the allure and pursuit of vain things such as wealth and power.&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

=====Symbols of Changing Times=====
The once elegant and decaying ship is a symbol of changing times.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; In a similar manner, Katie can be viewed as a symbolic representation of childhood from a more innocent era. Just as the ship decays to ruin from its glorious 1950’s splendor, Katie’s deteriorating childhood possessions and even Katie herself, are evocative symbols of changing times and eroding childhood innocence.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

== Dichotomy and Contrasts ==
Katie is in some ways a complicated paradox: She has been deeply hurt and traumatized by all she has experienced, but by the same token, she is still very much an innocent child – Even after 40 years of resilient growth as a character. Katie exhibits a wide array of emotions from timid youthful innocence and joy, to extreme terror, sadness, loneliness, and solemn distress.&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

====Characters====
Even under tremendously trying circumstances Katie demonstrates an incredible inner strength of character without compromising the sweet kind-hearted &quot;girly&quot; girl she really is. The fact that Katie remains true to herself in spite of all the evil and suffering she experienced transcends their eroding influence and distinguishes her from the other characters.&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

=====Ferriman=====
Ferriman is representative of Satan and the crafty deceptiveness of evil in destroying people’s lives. Katie and Ferriman are at completely opposite ends of the Christian spectrum – The demonic Ferriman is totally depraved and evil while Katie is sweet and innocent. Katie is motivated by love and compassion, Ferriman is motivated by hate and rage.&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt; &lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Epps=====
Although Katie and Epps are two completely dissimilar characters from different eras, they still grow quite close. Epps is &quot;a woman working in a man's world&quot;&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt; and appears somewhat tough and masculine on the exterior like a tomboy. Although not specifically stated, the story indicates that Epps is likely from a broken home. On the other hand, Katie comes from a close, traditional nuclear family. In contrast to Epps, Katie is more of a typical &quot;girly&quot; girl and is slightly timid, but possess a stout inner fortitude despite her diminutive and delicate appearance.&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

=====Francesca=====
The sultry Francesca lives the more representative life of a performing artist. Unlike Epps who feels the need to downplay her femininity, Francesca leans toward the other extreme and attempts to flaunt her femininity to her own advantage, even to the point of her own exploitation. Conversely, Katie is still a virtuous and modest little girl, who has yet to be tainted by any corrupting influences of the world.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Captain Murphy=====
Captain Murphy appears to be a strong imposing ship captain on the outside, however, when confronted with trials and tribulations, his perceived strength is shown to be superficial. This is evident by his emotional breakdown where he shuts off emotionally and withdraws to himself while attempting to drown his problems in alcohol. In contrast, Katie appears to be very delicate in both physical appearance and personality. However, despite her dainty appearance, Katie proves her true strength and fortitude from within through dealing with her pain and problems directly, instead of allow it to define her&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

====General Themes====
Light vs. darkness, good vs. evil, childhood innocence vs. wickedness, tragedy vs. triumph, simpler more wholesome times, vs. the complexity and loss of innocence of today.&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

== Conclusion ==
On the surface, Katie’s story is an emotional tale of heartache and tragedy – Something that no young person should ever have to endure. However, Katie’s story is also essential to the plot. &lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt; Through helping each other and being willing to put themselves at risk, Katie and Epps together achieve a triumph out of the tragedy and deliver a major blow to Ferriman and the forces of evil.  Although the struggle over the souls of people will continue, this victory of the human spirit gives hope even in the face of terrible suffering, evil and even death.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

“Katie is really a very sweet girl.  She's completely innocent…“ says Emily Jane Browning, who obviously became completely immersed in her character.  “…She's been hoping someone would come onto the ship to be her friend, so when Epps arrives she's very excited – they develop a real friendship.&quot;&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

”Emily did an amazing job,” Director Steve Beck enthuses. “She gave Katie a real complexity…she's not just a little girl caught up in a ghost story…”  Both Browning and Beck also added that underneath all her heartfelt emotions, Katie longs for vindication and justice.&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

The deeply symbolic and metaphorical aspects surrounding Katie’s story along with the close relationships she develops with others lends support to the notion that the film was originally intended to be more than just another Hollywood “slasher” horror. This is further substantiated by Katie’s tearful and heartfelt story – Emotionally poignant and touching drama not usually associated with a horror picture.&lt;ref&gt;http://www.dailyscript.com/scripts/ghost_ship_info.txt&lt;/ref&gt;

== References ==
===Footnotes===
{{Reflist|colwidth=30em}}

===Cited Texts===
{{refbegin}}

*{{cite book| last=Barclay | first=William | title=The Gospel of Mark |publisher=Westminster Press | location=Philadelphia, PA | isbn=0664213022 | year=1975}}
*{{cite book| last=Barton |first=John | title=The Oxford Bible Commentary |publisher=Oxford University Press |location=New York, NY  | isbn=0-19-875500-7 | year=2001}}
*{{cite book| last=Boettner |first=Loraine | title=Roman Catholicism |publisher=Presbyterian &amp; Reformed Publishing Co. |location=Phillipsburg, NJ |isbn=0875521304 | year=1985}}
*{{cite book| last=Buttrick |first=George | title=The Interpreter's Bible vol. 12 |publisher=Abingdon Press |location=Nashville, TN |asin=B000HTP248 | year=1957}}
*{{cite book |title=Collins English Dictionary - Complete &amp; Unabridged 10th Edition  |date=2009 | publisher=Harper Collins |location=New York, NY}}
*{{cite book| last=Dummelow |first=J.R. | title=Commentary on the Whole Bible |publisher=Macmillian Publishing Co. |location=New York, NY | year=1936}}
*{{cite book| last=Gaebelein |first=Frank | title=The Expositors Bible Commentary vol. 1 |publisher=Zondervan |location=Grand Rapids, MI |isbn=0310364302 | year=1979}}
*{{cite book| last=Gauding |first=Madonna | title=The Signs and Symbols Bible |publisher=Sterling Publishing Co. |location=New York, NY |isbn=1402770049| year=2009}}
*{{cite book| last=Jamieson |first=Fausset | title=Commentary on the Whole Bible  |publisher=Zondervan |location=Grand Rapids, MI| asin=B004BCQP8O | year=1971}}
*{{cite book| last=Keck |first=Leander | title=New Interpreter's Bible vol. 9 |publisher=Abingdon Press |location=Nashville, TN |isbn=0687278228 | year=2002}}
*{{cite book| title=Life Application Bible - New International Version |publisher=Tyndale House Publishers, Inc. |location=Wheaton, IL | lccn=90-71553| year=1991}}
*{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN |isbn=0840752954 | year=1983}}
*{{cite book| last=Wall |first=Robert | title=The New Interpreter's Bible vol. 10 |publisher=Abingdon Press |location=Nashville, TN |isbn=0687278236 | year=2002}}
*{{cite book| last=Whalen |first=John | title=New Catholic Encyclopedia Vol. 11 |publisher=Catholic University of America |location=Washington D.C. | lccn=66-22292 | year=1967}}

{{refend}}

== External links ==
* [http://www.dailyscript.com/scripts/ghost_ship_info.txt/ Daily Script_Ghost Ship]
* [http://www.imdb.com/title/tt0288477/ Internet Movie Database_Ghost Ship 2002]
* [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]
* [http://www.rottentomatoes.com/m/ghost_ship/ Rotten Tomatoes movie review_Ghost Ship]
[[Category:Analysis]]
[[Category:Film]]</text>
      <sha1>np2a533cucx2oqglbfc1x4kn4uzknq8</sha1>
    </revision>
    <revision>
      <id>2807850</id>
      <parentid>2807846</parentid>
      <timestamp>2026-05-06T21:47:19Z</timestamp>
      <contributor>
        <username>Tola73</username>
        <id>3020076</id>
      </contributor>
      <minor />
      <comment>/* Significance */ Changed one word for better readability.</comment>
      <origin>2807850</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="27930" sha1="2q96toanqk3grl0dgftqpm7v9q49eac" xml:space="preserve">'''Katherine &quot;Katie&quot; Harwood''' is a fictional character in the 2002 film ''Ghost Ship'' where an innocent young girl goes on a sea voyage of a lifetime, only to be caught up in a living nightmare aboard the ill-fated ocean liner. In the film, Katie is the supporting deuteragonist to the main character (Maureen Epps) and stands in stark contrast to the completely evil and demonic antagonist (Jack Ferriman). In many regards, Katie is just as much of a heroine as Maureen Epps for enduring unfathomable suffering and risking the wrath of Jack Ferriman through her unyielding efforts to save the souls and lives of others on the ship.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; Katie is portrayed by a young [[w:Emily Browning|Emily Jane Browning]].

== Significance ==
Katie is an iconic representation of childhood from an earlier era, at a time when life was simpler, and childhood more innocent.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; &lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt; She stands out in a coarse mixed reviewed horror film as a compelling, distinctive and enduring character who subverts common horror child tropes.&lt;ref&gt;https://michellepatterson.net/2018/03/23/ghostship-film-review/Michelle Patterson Publications&lt;/ref&gt;&lt;ref&gt;https://moviefilmreview.com/186187/movie-review-of-ghost-ship-2002?utm_source=chatgpt.com/ Cal Knox Movie Review&lt;/ref&gt;&lt;ref&gt;https://lmariewood.com/2025/05/23/horror-tropes-when-to-use-them-and-when-to-subvert-them/ L. Marie Wood; Horror Tropes: When to Use Them and When to Subvert Them&lt;/ref&gt; With the Shout! Factory 20-year anniversary Blue-ray rerelease of ''Ghost Ship,'' Katie is made the new face of the film in the promotional artwork, elevating a morally intact child to iconic status; something very few horror children achieve, and even more so as a tragic and sympathetic figure without villainy.&lt;ref&gt;https://www.blu-ray.com/news/?id=27285/ Shout! Factory Ghost Ship 20-year anniversary collector’s edition Blu-ray disc&lt;/ref&gt;

== Synopsis ==
In May of 1962, an endearing Katie waves goodbye to her grandparents in [[Europe]] and journeys solo aboard the exquisite ''Antonia Graza'' on an exclusive cruise to rejoin her family in New York City. Since Katie is the only child onboard, she receives special care and attention. In describing her voyage Katie states, ”The whole ship was my playground. I was the only child onboard, but the ship’s purser and captain took special care of me. I felt so safe and happy with them.”&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

However, after the Antonia Graza rescues a lone survivor and cargo from a sinking ship, Katie’s exciting journey suddenly takes a horrendous turn for the worse. Unbeknownst to Katie and her shipmates, the single survivor (Jack Ferriman) is literally a demonic henchman for Satan set on destroying lives and collecting souls. Ferriman informs select members of the crew about the millions in gold recovered from his sinking ship, and influences them to launch an elaborate plot to seize control of the Graza and the gold, by killing everyone onboard. Although several crewmembers and passengers try to save her life from the murderous conspirators, Katie is eventually caught and tragically hanged to death, with her body concealed behind the partitioning door of her cabin.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

Instead of ascending to her rightful place in heaven, Katie’s spirit is trapped on the ship by demonic forces along with the entire compliment of murdered passengers and crew. However, since Katie is still a young child, her soul is completely innocent and therefore beyond the influence and control of the demonic Ferriman. Over the next 40 years, the evil Ferriman along with the help of the wicked “marked” souls, try to use the ship as a trap to destroy unsuspecting lives, and collect a quota of souls for hell. Despite being the only flicker of good on the ship, Katie bravely opposes Ferriman and the evil spirits, and attempts to warn and save the lives of anyone who has the misfortune of being lured aboard the condemned vessel. Concerning these things, Katie says, ““Without the mark, Jack can’t control me and because of this, he hates me most of all; scaring me at every turn and chasing me away when I try to warn those who come here.”&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

When Maureen Epps and her ship salving crew are lured aboard the Graza by Ferriman, Katie attempts to warn them about the dangerous ship. Katie has very perceptive eyes, and finds Epps to be more open to the truth than her cohorts. Katie tries to leave Epps messages and clues, and even appears to her on several occasions. When Katie senses that Ferriman has sabotaged the salvage crew’s tugboat, she boldly tries to warn the crew, but is forcibly carried away by Ferriman. Epps witnesses Katie’s warning and sets off in search for the mysterious little girl.

Epps locates Katie’s cabin and comes face to face with her remains and her spirit. Katie allows Epps to have her cherished heart-shaped locket, and then proceeds to speak directly about the ship being a demonic trap. However, before Katie can finish explaining, she is overheard by an invisible Ferriman and lets out a frightened scream as he removes her from the cabin. Undeterred, Katie returns and tries to help Epps and her remaining crewmembers escape. Endowed with supernatural power from above, Katie transports Epps back to May 21, 1962 through her memories and reveals to her the horrifying events that led to everyone’s death, including her own. Furthermore, Katie reveals the true identity of Jack Ferriman and his goal to use the ship as a conduit to collect souls for Satan.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

After experiencing Katie’s heartbreaking vision, Epps decides to risk almost certain death by destroying the cursed ship with explosives. However, Katie helps Epps escape the rapidly sinking vessel, while emancipated souls rise from the ship. Katie stays faithfully by Epps and gives her a grateful smile as her spirit ascends toward heaven.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

== Religious &amp; Symbolic Implications ==
At its core, the story is deeply religious and the plot hinges on the concept of a young girl trying to save the lives (and ultimately souls) of others from demonic forces.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Metaphorical Murals=====
The religious implications and [[Christian symbolism]] surrounding Katie’s story are rich. &lt;ref&gt;http://www.worsleyschool.net/socialarts/symbolism/page.html&lt;/ref&gt; Although only lightly touched on in the final version of the film, the most obvious religious connections come through the [[Gustave Dore]] inspired murals displayed throughout the ship. The murals are depictions based on Dante Alighieri’s Divine Comedy and The Inferno. The metaphorical murals hint at the struggle between Katie and Ferriman over the lives and souls on the ship.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; Lead actress Julianna Margulies says, “Katie is like Virgil in Dante’s Inferno. Margulies explains that Katie is like a “little guide” trying to lead others through hell and then safely onto the other side.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; As in the mural of Charon, Ferriman is literally a “ferry man” using a boat in the attempt to ferry souls to hell and eternal damnation.  Conversely, Katie opposes Ferriman by trying to protect and warn all unsuspecting visitors and sharing the truth about the demonic ship.

=====Marked Souls=====
An additional concept presented in the plot are the “marked” souls. Like the murals in the story, the souls of the sinful and the “lost”&lt;ref&gt;{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN |page=563 | year=1983}}&lt;/ref&gt; are marked with a hooked-shaped indention on their hand. Katie says the mark is “a sign of their sins” and that they are bound to the will of Satan and his demons, in addition to eventually sharing in their same flaming judgment.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; However, not all the murals are somber. The depiction of angels defeating the demons and casting them down foreshadows the eventual defeat of Ferriman and ultimately all evil.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====The Dove and Locket=====
Another subtle, but implicit [[Christian symbolism]] is Katie’s heart-shaped locket with the image of a raised dove in flight.&lt;ref&gt;{{cite book| last=Barton |first=John | title=The Oxford Bible Commentary |publisher=Oxford University Press |location=New York, NY  | year=2001 |page=383}}&lt;/ref&gt; In Christian tradition, this type of dove can symbolize innocence, purity, faith, hope, God’s peace, God’s presence through the [[Holy Spirit (Christianity)|Holy Spirit]] and God’s guidance and deliverance.&lt;ref&gt;http://www.newadvent.org/cathen/05144b.htm&lt;/ref&gt; The dove also indicated God’s bestowal of fortitude necessary to bear suffering and death&lt;ref&gt;{{cite book| last=Gauding |first=Madonna | title=The Signs and Symbols Bible |publisher=Sterling Publishing Co |location=New York, NY  | year=2009 |page=82}}&lt;/ref&gt;. In this sense, the remains of the dead dove on the bridge of the ship, foreshadows Katie’s death, but at the same time indicates that there could still be hope.  When Epps puts on Katie’s locket, it implies that Epps now has a renewed heart, or possibly even salvation as stated in Bible scripture. In this manner, the locket could also represent that Epps’ soul is not &quot;marked&quot;.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Parallel of an Innocent and Sacrificial Life=====
A striking parallel could also be drawn between Katie and Jesus Christ. In the bible, Christ lives a completely innocent life, but is hung on a cross to sacrificially die for the sins of all humanity.&lt;ref&gt;{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN  | year=1983 |pages=2228-2229}}&lt;/ref&gt; Christ tastes the sting of death and hell in order to offer salvation to all who will put their faith in Him.&lt;ref&gt;{{cite book| last=Barton |first=John | title=The Oxford Bible Commentary |publisher=Oxford University Press |location=New York, NY  | year=2001 |page=966}}&lt;/ref&gt; Similarly, innocent Katie is hung by a rope behind a dividing door, but through her death she is able to lead others to &quot;salvation&quot; and safety. Because of her young age and innocent death, Katie is the only voice of hope for all the suppressed souls held captive on the demonic ship. Katie also endures small tastes of hell from Ferriman in her efforts to save others. However, Katie’s love and compassion for the lives of others is greater than her fear and pain, and in similarity to Christ, Katie rises victoriously to heaven in the end.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Forty Years and the Promised Land=====
40 years is a significant reoccurring number in the Christian faith.&lt;ref&gt;Collins English Dictonary http://dictionary.reference.com/browse/promised land&lt;/ref&gt; The [[Israelites]] endured 40 years of great trial and testing in the wilderness before they were able to enter their promised land of rest. In Christian tradition, the earthly life is often compared to a time of trial in the wilderness and the Promised Land a metaphor for heaven.&lt;ref&gt;Collins English Dictonary http://dictionary.reference.com/browse/promised land&lt;/ref&gt; Correspondingly, Katie has endured the “wilderness” and trials of the ship for 40 years, but as the result of Epps’ help she is inevitably freed to enter the “promised land” of heaven.

=====Concept of Free Will=====
Free will is both discussed and demonstrated in the story.&lt;ref&gt;{{cite book| last=Keck |first=Leander | title=The New Interpreter's  Bible vol. 9 |publisher=Abingdon Press |location=Nashville, TN  | year=2002 |pages=61-62}}&lt;/ref&gt; Ultimately, the crew members who die are led to destruction by their own sins and their selfish desire for greed, lust and power. Epps is spared the fate of her shipmates through listening to Katie’s warnings and humbly responding to the truth.&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

=====Elevated Status of Children=====
In the Christian bible God often uses the most humble and unlikely people to work through &lt;ref&gt;{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN  | year=1983 |page=2283}}&lt;/ref&gt; – Out of hundreds on the ship, Katie is the meekest, humblest and most unlikely to do anything significant, yet every soul on the ship ends up depending on her.&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page production notes]&lt;/ref&gt;

Katie is also reminiscent of Christ elevating the status of children when he gave them special attention and profoundly declared that unless one humbles them self and has the faith of a little child, they will not enter the kingdom of heaven. This is further illustrated when Christ took a lowly child and placed him in the midst of his followers and stated that only those with childlike faith will be the greatest in the kingdom of heaven.&lt;ref&gt;{{cite book| last=Barton |first=John | title=The Oxford Bible Commentary |publisher=Oxford University Press |location=New York, NY  | year=2001 |page=963}}&lt;/ref&gt; This concept is demonstrated by Epps' willingness to humble herself to Katie's level.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

The role of children in Christian scripture – Comparisons could be made between Katie and several unpretentious bible characters such as the lowly shepherd boy David who defeated the warrior Goliath and became king. Outwardly both young Katie and David appear to be nothing more that mere children, however, inwardly they possess vast unforeseen potential and a tremendous strength of character.&lt;ref&gt;{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN  | year=1983 |page=563}}&lt;/ref&gt; &lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Eternal Destinations=====
In the film, as with [[Christian soteriology]] there are two clear eternal destinations which souls go to after death; ether heaven or hell.  A heart of child-like faith will put one on the path to heaven, but a sinful life eventually leads to destruction.&lt;ref&gt;{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN  | year=1983 |page=1902}}&lt;/ref&gt;

=====The Destructiveness of Sinful Choices=====
One of the morals of the story is the corrupting dangers of pursuing riches. This theme also fits aptly within the religious concepts of the film.&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt; Sinful greed, lust, debauchery and selfish desire can influence ordinarily good people to do horribly evil things (1Timothy 6:9-10).&lt;ref&gt;{{cite book| title=Life Application Bible - New International Version |publisher=Tyndale House Publishers, Inc. |location=Wheaton, IL | page=2195 | year=1991}}&lt;/ref&gt; The choices made by the characters in the story illustrate this principle.

“Be sure your sins will find you out” – As the story goes, sinful activities have consequences and will eventually have to be accounted for, and the ultimate payment for sin in the end is both physical and spiritual death (Romans 6:20-23). &lt;ref&gt;{{cite book| title=Life Application Bible - New International Version |publisher=Tyndale House Publishers, Inc. |location=Wheaton, IL | page=2039 | year=1991}}&lt;/ref&gt; &lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

=====Loving Others before Ourselves=====
“Greater love has no one than this – To lay one’s life down for their friends.” Katie and Epps are both willing put themselves at risk to save lives. Through the process of trying to help others both Epps and Katie experience help for themselves.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

“Those who seek to save their life will lose it, but those who are willing to lose their life will save it.”&lt;ref&gt;{{cite book| title=Life Application Bible - New International Version |publisher=Tyndale House Publishers, Inc. |location=Wheaton, IL | page=1842 | year=1991}}&lt;/ref&gt; In this sense Epps' life is saved, because she places the needs of others first. Katie also inherits eternal life in heaven by risking herself to help others.

=====Purgatory=====
The concept of purgatory (Whalen p. 1034-1039)&lt;ref&gt;{{cite book| last=Whalen |first=John | title=New Catholic Encyclopedia Vol. 11|publisher=Catholic University of America |location=Washington, D.C.  | year=1983}}&lt;/ref&gt; is also mentioned when Katie states that the ship had become like a “prison” where souls are trapped among the living between heaven and hell.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

====Additional Symbolism====
=====Life as a Voyage on a Ship=====
Life itself is comparable to a voyage on ship...illustrating that our lives are like a ship that is bound for one of two possible destinations.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; In the context of Ghost Ship, the voyage ends in destruction (death and hell) unless we get off. In this sense, Katie pleads with us to get off, but that involves humbly changing our ways and giving up the allure and pursuit of vain things such as wealth and power.&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

=====Symbols of Changing Times=====
The once elegant and decaying ship is a symbol of changing times.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt; In a similar manner, Katie can be viewed as a symbolic representation of childhood from a more innocent era. Just as the ship decays to ruin from its glorious 1950’s splendor, Katie’s deteriorating childhood possessions and even Katie herself, are evocative symbols of changing times and eroding childhood innocence.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

== Dichotomy and Contrasts ==
Katie is in some ways a complicated paradox: She has been deeply hurt and traumatized by all she has experienced, but by the same token, she is still very much an innocent child – Even after 40 years of resilient growth as a character. Katie exhibits a wide array of emotions from timid youthful innocence and joy, to extreme terror, sadness, loneliness, and solemn distress.&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

====Characters====
Even under tremendously trying circumstances Katie demonstrates an incredible inner strength of character without compromising the sweet kind-hearted &quot;girly&quot; girl she really is. The fact that Katie remains true to herself in spite of all the evil and suffering she experienced transcends their eroding influence and distinguishes her from the other characters.&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

=====Ferriman=====
Ferriman is representative of Satan and the crafty deceptiveness of evil in destroying people’s lives. Katie and Ferriman are at completely opposite ends of the Christian spectrum – The demonic Ferriman is totally depraved and evil while Katie is sweet and innocent. Katie is motivated by love and compassion, Ferriman is motivated by hate and rage.&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt; &lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Epps=====
Although Katie and Epps are two completely dissimilar characters from different eras, they still grow quite close. Epps is &quot;a woman working in a man's world&quot;&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt; and appears somewhat tough and masculine on the exterior like a tomboy. Although not specifically stated, the story indicates that Epps is likely from a broken home. On the other hand, Katie comes from a close, traditional nuclear family. In contrast to Epps, Katie is more of a typical &quot;girly&quot; girl and is slightly timid, but possess a stout inner fortitude despite her diminutive and delicate appearance.&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

=====Francesca=====
The sultry Francesca lives the more representative life of a performing artist. Unlike Epps who feels the need to downplay her femininity, Francesca leans toward the other extreme and attempts to flaunt her femininity to her own advantage, even to the point of her own exploitation. Conversely, Katie is still a virtuous and modest little girl, who has yet to be tainted by any corrupting influences of the world.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

=====Captain Murphy=====
Captain Murphy appears to be a strong imposing ship captain on the outside, however, when confronted with trials and tribulations, his perceived strength is shown to be superficial. This is evident by his emotional breakdown where he shuts off emotionally and withdraws to himself while attempting to drown his problems in alcohol. In contrast, Katie appears to be very delicate in both physical appearance and personality. However, despite her dainty appearance, Katie proves her true strength and fortitude from within through dealing with her pain and problems directly, instead of allow it to define her&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

====General Themes====
Light vs. darkness, good vs. evil, childhood innocence vs. wickedness, tragedy vs. triumph, simpler more wholesome times, vs. the complexity and loss of innocence of today.&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

== Conclusion ==
On the surface, Katie’s story is an emotional tale of heartache and tragedy – Something that no young person should ever have to endure. However, Katie’s story is also essential to the plot. &lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt; Through helping each other and being willing to put themselves at risk, Katie and Epps together achieve a triumph out of the tragedy and deliver a major blow to Ferriman and the forces of evil.  Although the struggle over the souls of people will continue, this victory of the human spirit gives hope even in the face of terrible suffering, evil and even death.&lt;ref&gt;Ghost Ship DVD (2003) - Extra features and behind the scenes&lt;/ref&gt;

“Katie is really a very sweet girl.  She's completely innocent…“ says Emily Jane Browning, who obviously became completely immersed in her character.  “…She's been hoping someone would come onto the ship to be her friend, so when Epps arrives she's very excited – they develop a real friendship.&quot;&lt;ref&gt;[http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

”Emily did an amazing job,” Director Steve Beck enthuses. “She gave Katie a real complexity…she's not just a little girl caught up in a ghost story…”  Both Browning and Beck also added that underneath all her heartfelt emotions, Katie longs for vindication and justice.&lt;ref&gt; [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]&lt;/ref&gt;

The deeply symbolic and metaphorical aspects surrounding Katie’s story along with the close relationships she develops with others lends support to the notion that the film was originally intended to be more than just another Hollywood “slasher” horror. This is further substantiated by Katie’s tearful and heartfelt story – Emotionally poignant and touching drama not usually associated with a horror picture.&lt;ref&gt;http://www.dailyscript.com/scripts/ghost_ship_info.txt&lt;/ref&gt;

== References ==
===Footnotes===
{{Reflist|colwidth=30em}}

===Cited Texts===
{{refbegin}}

*{{cite book| last=Barclay | first=William | title=The Gospel of Mark |publisher=Westminster Press | location=Philadelphia, PA | isbn=0664213022 | year=1975}}
*{{cite book| last=Barton |first=John | title=The Oxford Bible Commentary |publisher=Oxford University Press |location=New York, NY  | isbn=0-19-875500-7 | year=2001}}
*{{cite book| last=Boettner |first=Loraine | title=Roman Catholicism |publisher=Presbyterian &amp; Reformed Publishing Co. |location=Phillipsburg, NJ |isbn=0875521304 | year=1985}}
*{{cite book| last=Buttrick |first=George | title=The Interpreter's Bible vol. 12 |publisher=Abingdon Press |location=Nashville, TN |asin=B000HTP248 | year=1957}}
*{{cite book |title=Collins English Dictionary - Complete &amp; Unabridged 10th Edition  |date=2009 | publisher=Harper Collins |location=New York, NY}}
*{{cite book| last=Dummelow |first=J.R. | title=Commentary on the Whole Bible |publisher=Macmillian Publishing Co. |location=New York, NY | year=1936}}
*{{cite book| last=Gaebelein |first=Frank | title=The Expositors Bible Commentary vol. 1 |publisher=Zondervan |location=Grand Rapids, MI |isbn=0310364302 | year=1979}}
*{{cite book| last=Gauding |first=Madonna | title=The Signs and Symbols Bible |publisher=Sterling Publishing Co. |location=New York, NY |isbn=1402770049| year=2009}}
*{{cite book| last=Jamieson |first=Fausset | title=Commentary on the Whole Bible  |publisher=Zondervan |location=Grand Rapids, MI| asin=B004BCQP8O | year=1971}}
*{{cite book| last=Keck |first=Leander | title=New Interpreter's Bible vol. 9 |publisher=Abingdon Press |location=Nashville, TN |isbn=0687278228 | year=2002}}
*{{cite book| title=Life Application Bible - New International Version |publisher=Tyndale House Publishers, Inc. |location=Wheaton, IL | lccn=90-71553| year=1991}}
*{{cite book| last=Towns |first=Elmer | title=Liberty Bible Commentary |publisher=Thomas Nelson Inc |location=Nashville, TN |isbn=0840752954 | year=1983}}
*{{cite book| last=Wall |first=Robert | title=The New Interpreter's Bible vol. 10 |publisher=Abingdon Press |location=Nashville, TN |isbn=0687278236 | year=2002}}
*{{cite book| last=Whalen |first=John | title=New Catholic Encyclopedia Vol. 11 |publisher=Catholic University of America |location=Washington D.C. | lccn=66-22292 | year=1967}}

{{refend}}

== External links ==
* [http://www.dailyscript.com/scripts/ghost_ship_info.txt/ Daily Script_Ghost Ship]
* [http://www.imdb.com/title/tt0288477/ Internet Movie Database_Ghost Ship 2002]
* [http://ghostshipmovie.warnerbros.com/production_notes.html?page=2/ Ghost Ship official movie page_production notes]
* [http://www.rottentomatoes.com/m/ghost_ship/ Rotten Tomatoes movie review_Ghost Ship]
[[Category:Analysis]]
[[Category:Film]]</text>
      <sha1>2q96toanqk3grl0dgftqpm7v9q49eac</sha1>
    </revision>
  </page>
  <page>
    <title>Linux System programming in plain view</title>
    <ns>0</ns>
    <id>136794</id>
    <revision>
      <id>2807820</id>
      <parentid>2807693</parentid>
      <timestamp>2026-05-06T15:50:03Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* File System */</comment>
      <origin>2807820</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3197" sha1="cg6btpo5kbbodtzb9ukk56avuci95hu" xml:space="preserve">This course belongs to the [[Electrical &amp; Computer Engineering Studies]]

== Introduction ==
* Introduction ([[Media:SysP.Intro.20161128.pdf|pdf]])

== File System ==
* File System ([[Media:SysP.FileSystem.20251023.pdf|pdf]])
* File Pointer ([[Media:SysP..FilePointer.20161103.pdf|pdf]])
* System Calls ([[Media:SysP.File.SysCall.20161128.pdf|pdf]])
* File IO ([[Media:SysP.FileIO.20251023.pdf|pdf]])


* Copilot: File System ([[Media:glibcFileSystem.20251029-2.pdf|pdf]])
* Copilot: File Buffer ([[Media:glibcFileBuffer.20251025-2.pdf|pdf]])
* Copilot: File IO ([[Media:glibcFileIO.20251025-2.pdf|pdf]])
* Copilot: File Permission ([[Media:glibcFilePerm.20260121.pdf|pdf]])
* Copilot: File Control ([[Media:CP.FileCntl.20260428.pdf|pdf]], [[Media:CP.FileCntl.A.20260506.pdf|A]], [[Media:CP.FileCntl.B.20260504.pdf|B]], [[Media:CP.FileCntl.C.20260501.pdf|C]])

&lt;br&gt;
&lt;br&gt;

== Process ==
* Process  ([[Media:SysP.Process.20251120.pdf|pdf]])
* Fork  ([[Media:SysP.Fork.20251126.pdf|pdf]])


* Copilot: Process Information ([[Media:glibc.Process.1Info.20251101.pdf|pdf]])
* Copilot: Process Control ([[Media:glibc.Process.2Control.20251103.pdf|pdf]])
* Copilot: Process Execution ([[Media:glibc.Proc.3Exec.20251105.pdf|pdf]])
* Copilot: Process Fork ([[Media:glibc.Proc.4Fork.20251106.pdf|pdf]])
* Copilot: Process Context Switching ([[Media:glibc.Proc.5Context.20251107.pdf|pdf]])
* Copilot: Process Exec family of functions ([[Media:glibc.Proc.6ExecCall.20251112.pdf|pdf]])
* Copilot: Process Wait family of functions ([[Media:glibc.Proc.7WaitCall.20251112.pdf|pdf]])
* Copilot: Process Exit ([[Media:glibc.Proc.8Exit.20251113.pdf|pdf]])

&lt;/br&gt;

== Inter Process Communication==

=== Signal ===
* Signal  ([[Media:SysP.7.A.Signal.20121206.pdf|pdf]])


* Copilot: Signal 1. Alarm ([[Media:glibc.Signal.Alarm.20251201.pdf|pdf]])
* Copilot: Signal 2. Other Functions ([[Media:glibc.Signal.2Other.20251205.pdf|pdf]])

&lt;/br&gt;

=== Pipe ===

* Pipe ([[Media:SysP.3.A.IPC.20121115.pdf|pdf]])


* Copilot: Pipe 1. A Special File ([[Media:glibc.Pipe.File.20260307.pdf|pdf]])

&lt;/br&gt;

=== System V IPC ===

* Message Queue  ([[Media:SysP.5.A.MessageQ.20121213.pdf|pdf]])

* Shared Memory  ([[Media:SysP.8.A.SharedMem.20121227.pdf|pdf]])

* Semaphore  ([[Media:SysP.6.A.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

* Copilot: Message Queue ([[Media:glibc.MessageQ.20251202.pdf|pdf]])
* Copilot: Shared Memory ([[Media:glibc.SharedMem.20251203.pdf|pdf]])
* Copilot: Semaphore ([[Media:glibc.Semaphore.20251215.pdf|pdf]])

&lt;/br&gt;

=== Socket ===

* Socket  ([[Media:SysP.4.A.Socket.20121122.pdf|pdf]])

&lt;/br&gt;

== Thread ==
* POSIX thread (pthread)  ([[Media:SysP.9.A.Pthread.20130225.pdf|pdf]])

==External links==
* [http://www.tldp.org/LDP/tlk/tlk.html The Linux Kernel]
* [http://www.tldp.org/LDP/lpg/lpg.html The Linux Programmer's Guide]
* [http://www.cs.cf.ac.uk/Dave/C/ Programming in C - UNIX System Calls and Subroutines using C.]
* [http://www.cs.cmu.edu/afs/cs/academic/class/15492-f07/www/pthreads.html POSIX thread (pthread) libraries]
* [https://computing.llnl.gov/tutorials/pthreads/#Thread POSIX Threads Programming]

[[Category:Linux]]
[[Category:Computer programming]]
[[Category:C programming language]]</text>
      <sha1>cg6btpo5kbbodtzb9ukk56avuci95hu</sha1>
    </revision>
  </page>
  <page>
    <title>Understanding Arithmetic Circuits</title>
    <ns>0</ns>
    <id>139384</id>
    <revision>
      <id>2807808</id>
      <parentid>2807640</parentid>
      <timestamp>2026-05-06T13:32:03Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Adder */</comment>
      <origin>2807808</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3135" sha1="1bkc8bhsjza6mix9z5bc530kv2qk1g7" xml:space="preserve">
== Adder ==

* Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]]  )
 
{| class=&quot;wikitable&quot;
|-
!  Adder type      !! Overview !! Analysis !! VHDL Level Design  !! CMOS Level Design  
|-
| '''1. Ripple Carry Adder'''
|| [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]||
|| [[Media:Adder.rca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]]
|-
| '''2. Carry Lookahead Adder'''
|| [[Media:VLSI.Arith.1.A.CLA.20260109.pdf|org]], [[Media:VLSI.Arith.2A.CLA.20260506.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260505.pdf|B]] ||
|| [[Media:Adder.cla.20140313.pdf|pdf]]||
|-
| '''3. Carry Save Adder''' 
|| [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]||
|| ||
|-
|| '''4. Carry Select Adder''' 
|| [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]||
|| || 
|-
|| '''5. Carry Skip Adder'''
|| [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]||
|| 
|| [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]]
|- 
|| '''6. Carry Chain Adder'''
|| [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]||
|| [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]],  [[Media:Adder.cca.20140313.pdf|pdf]]
|| [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]]
|- 
|| '''7. Kogge-Stone Adder'''
|| [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]||
|| [[Media:Adder.ksa.20140409.pdf|pdf]]||
|-
|| '''8. Prefix Adder'''
|| [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]||
|| || 
|-
|| '''9.1 Variable Block Adder'''
|| [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]||
|| || 
|-
|| '''9.2 Multi-Level Variable Block Adder'''
|| [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]||
|| || 
|}
&lt;/br&gt;
 
=== Adder Architectures Suitable for FPGA === 
* FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]])
* FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]])
* FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]])
* FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]])
* Carry-Skip Adder
&lt;/br&gt;

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

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

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

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

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

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

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

[[Category:Digital Circuit Design]]
[[Category:FPGA]]</text>
      <sha1>1bkc8bhsjza6mix9z5bc530kv2qk1g7</sha1>
    </revision>
  </page>
  <page>
    <title>Complex analysis in plain view</title>
    <ns>0</ns>
    <id>171005</id>
    <revision>
      <id>2807814</id>
      <parentid>2807647</parentid>
      <timestamp>2026-05-06T13:52:12Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Geometric Series Examples */</comment>
      <origin>2807814</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5296" sha1="mws3llh6ls4ed8ylx11n4b7w1niewz1" xml:space="preserve">Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers &lt;math&gt;x+iy&lt;/math&gt;, where &lt;math&gt;i=\sqrt{-1}&lt;/math&gt;, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other &lt;math&gt;L&lt;/math&gt;-functions, modular forms, elliptic functions, etc. &lt;blockquote&gt;The shortest path between two truths in the real domain passes through the complex domain.  — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]&lt;/blockquote&gt;In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}}

==''' Complex Functions '''== 
* Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]])

* Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]])

* Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]])

'''Complex Function Note'''
: 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]])

: 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]])

: 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]])

==''' Complex Integrals '''== 
* Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]])

==''' Complex Series '''== 
* Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]])

==''' Residue Integrals '''==
* Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) 

==='''Residue Integrals Note'''===
* Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]])
* Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]])
* Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]])
* Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]])

=== Laurent Series and the z-Transform Example Note === 
* Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]])

====Geometric Series Examples====

* Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) 

* Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) 

* Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) 

* Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) 

* Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) 

* Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260506.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) 

* Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) 

* Double Pole Case

:- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) 

:- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]])

====The Case Examples====
* Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]])
* Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]])
* Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]])
* Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]])
* Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]])
* Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]])

==''' Conformal Mapping '''== 
* Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]])

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

[[Category:Complex analysis]]</text>
      <sha1>mws3llh6ls4ed8ylx11n4b7w1niewz1</sha1>
    </revision>
  </page>
  <page>
    <title>The necessities in Filter Theory</title>
    <ns>0</ns>
    <id>199550</id>
    <revision>
      <id>2807822</id>
      <parentid>2807700</parentid>
      <timestamp>2026-05-06T16:34:10Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Sample Processing Methods */</comment>
      <origin>2807822</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1524" sha1="tpbexmhj7xsmwyme4vi5kgfd718qi9j" xml:space="preserve">==''' Background '''==
=== Bode plot ===

See [http://lpsa.swarthmore.edu/Bode/Bode.html swarthmore]
&lt;/br&gt;

=== OP Amp ===
Overview ([[Media:OPAmp.A.1.20151203.pdf  |pdf]])
 
See [http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/opampcon.html#c1 Hyperphysics]
&lt;/br&gt;

==''' Analog Filter Analysis (Continuous Time) '''==
=== First Order Filters ===
&lt;/br&gt;

=== Second Order Filters ===
&lt;/br&gt;

==''' Digital Filter Analysis (Discrete Time) '''==

=== Sample Processing Methods  ===
* Tapped Delays ([[Media:Sample.TappedDelay.20260506.pdf  |A.pdf]])
* Programming Considerations
* Circular Buffers

=== FIR Filter Realizations ===
* Direct Form FIR Filter 
* Canonical Form FIR Filter
* Cascade Form FIR Filter

=== IIR Filter Realizations ===
* Direct Form IIR Filter ([[Media:IIR.DirectForm.20231209.pdf  |A.pdf]])
* Canonical Form IIR Filter
* Cascade Form IIR Filter 
&lt;/br&gt;

=== FIR (Finite Impulse Response) Filters ===
* Block Processing Methods
* Sample Processing Methods
* Window Method
* Kaiser Window
* Frequency Sampling Method
&lt;/br&gt;

=== IIR (Infinite Impulse Response) Filters ===
* Bilinear Transform
* 1st Order Lowpass and Highpass Filters
* 2nd Order Lowpass and Highpass Filters
* Parametric Equalizer Filters
* Comb Filters
* High Order Filters
&lt;/br&gt;


=== Example Octave Codes for Digital Filters === 

==== Octave Functions for Filters ====
* Octave Functions for Filters ([[Media:Octave.1.Function.1.A.20180219.pdf  |A.pdf]]) 


&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
      <sha1>tpbexmhj7xsmwyme4vi5kgfd718qi9j</sha1>
    </revision>
  </page>
  <page>
    <title>Algebra 1/Unit 1: Introduction To Algebra</title>
    <ns>0</ns>
    <id>217152</id>
    <revision>
      <id>2807863</id>
      <parentid>2782433</parentid>
      <timestamp>2026-05-07T09:14:30Z</timestamp>
      <contributor>
        <username>~2026-27598-14</username>
        <id>3070872</id>
      </contributor>
      <origin>2807863</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21661" sha1="0frw34lye6k0xlgz0135kjc8dzjeac5" xml:space="preserve">[[File:Quadratic formula.svg|thumb|right|Algebra (quadratic formula)]]
{{mathematics}}
{{secondary education}}
{{lesson}}
{{complete}}

'''Algebra''' (from the Arabic word &quot;al-jabr&quot; (الجبر), meaning &quot;reunion of broken parts&quot;) can feel like quite a complicated language of mathematics. Anyone who masters the arts of Algebra is a true genius! This week, we will get into what Algebra is, and some warm ups (on arithmetic). Even though this may seem pointless, it is IMPORTANT that you review through these warm ups and get comfortable in solving them consistently to lay a strong foundation for understanding larger topics later on.

Without further do, let's dig right into this!
==Algebra==
===What is Algebra?===
[[File:AlgebraJournalWork11-14-16.jpg|thumb|left|You might have to do this much work for a small answer!]]
[[File:WhatwejustdidinthisproblemALGEBRA.PNG|thumb|right|This, summed up algebraically, is what we just did]]
A broad part of mathematics. In summary, Algebra solves for values that are not known yet. Such as □ - 4 = 6... instead of this empty box, we replace that with something known as a '''variable''', which is a letter that is used for something we do not know yet (of it's value). 

So, for the problem: □ - 4 = 6... we will have to do some math! We need to add positive 4, to the negative 4 on the left handside of the equation. When you add a number to another number (in which the solution becomes 0), you have cancelled out that number. But, you cannot just cancel out 4, and be done with it! You have to also add 4 to the number 6 on the righthand side of the equal sign, which brings out a golden rule in Algebra: '''To keep the balance, what we do to one side of the &quot;=&quot; we should also do to the other side!&quot;. ''

Now... once you have added 6 + 4 (which is 10), you should get □ = 10.

Great! You've done what's called &quot;solving for x&quot;, but in this case, it's an empty box... well guys, you're going to have to throw out that box, because it will be represented by a variable (a letter, from a - z). The letter that is commonly used for variables is x (and the reason for this  dates back to the origin of Algebra itself; Muhammad ibn Musa al-Khwarizmi, one of the main &quot;founders&quot; of Algebra you could say, used to call the unknown or &quot;box&quot; in our example, &quot;'''shay'''&quot;. &quot;'''Shay&quot;''' comes from the Arabic word '''شَيْء''', which essentially means &quot;thing&quot;. When Al-Khwarizmi's works were translated to Latin in medieval Spain, &quot;shay&quot; was translated as '''&quot;xay&quot;,''' since the letter x was pronounced as sh in Spain. Later on, this word &quot;'''xay&quot;''' got abbreviated to &quot;'''x&quot;''' to repesent the symbol of the unkown. Source/For more information, check out this [https://www.pbs.org/empires/islam/innoalgebra.html PBS] page). Brief history lesson aside, it's interesting to learn why &quot;x&quot; is the default norm when solving for the unkown in Algebra! '''Importantly, know that the unknown can take on any letter or variable name, not just x (i.e. could be a, b, c, y, even words if you wanted, etc.). Just remember what that variable is representing (the unknown value in this case).''' 

So instead of □ = 10, it's actually '''x = 10'''.

===== Sample problems of ''solving for x'' =====
&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{''x'' − 9 = 20
|type=&quot;{}&quot;}
''x''={ 29_3 }

{''x'' − 3 = 6
|type=&quot;{}&quot;}
''x''={ 9_3 }

{''x'' + 5 = 15
|type=&quot;{}&quot;}
''x''={ 10_3 }

{''x'' + 17 = 23
|type=&quot;{}&quot;}
''x''={ 6_3 }

{4''x'' = 12
|type=&quot;{}&quot;}
''x''={ 3_3 }

{''x''/2 = 0.5
|type=&quot;{}&quot;}
''x''={ 1_3 }

{''x''/50 = 2
|type=&quot;{}&quot;}
''x''={ 100_3 }

{''x''/9 = 5
|type=&quot;{}&quot;}
''x''={ 45_3 }
&lt;/quiz&gt;

Seems simple, huh? Well, it will get complicated, which is why it is important for you to do some review of your arithmetic! Let's dig into that...

==Arithmetic==
[[File:Multiply 4 bags 3 marbles.svg|thumb|right|4 x 3 = 12 (multiplication)]]
'''Arithmetic''' has to deal with elementary/basic levels of math, such as division, multiplication, subtraction, and addition. Basically, just working with numbers.

This SHOULD be a level familiar with you. If you are not familiar with arithmetic math/rules, then PLEASE review through Arithmetic, as you won't survive even the 1st step of Algebra. Trust me, the basics are THAT important.
===Fractions===
[[File:Cake quarters.svg|thumb|left|A Cake with fractions]]
'''Fractions''' (from Latin ''fractus'', &quot;broken&quot;) are parts of a whole. On the left side in the image of the cake, there is only &lt;math&gt;3/4&lt;/math&gt;'s of the cake showing, the other &lt;math&gt;1/4&lt;/math&gt; has been eaten/taken away.

The number, ''3'', in &lt;math&gt;3/4&lt;/math&gt;, is what is known as a '''numerator''' (Numerator: Number at the top, tells us of how much of the number is being talked about/being used). The number, ''4'', in &lt;math&gt;3/4&lt;/math&gt;, is what is known as a '''denominator''' (Denominator: Number showing the all time total).
;
;Simplest form/reduced form
A reduced form of a fraction is a fraction that cannot be divided by any number other than 1, and the denominator is greater than 1. So &lt;math&gt;2/4&lt;/math&gt; is NOT in simplest form, since we can divide 2 and 4, by 2... which results in the following number: &lt;math&gt;1/2&lt;/math&gt;. Though, not every fraction can be divided by 2, there are fractions, such as: &lt;math&gt;5/35&lt;/math&gt;, &lt;math&gt;7/21&lt;/math&gt;, and &lt;math&gt;30/5&lt;/math&gt;. The two first fractions are not divisible by 2, and &lt;math&gt;30/5&lt;/math&gt; can not be divided by 2 on both sides, but only on &lt;math&gt;30&lt;/math&gt;. It's important to simplify as if you were in a test, your teacher will mark your problems as incorrect if you didn't simplify your fractions. Keep in mind that simplifying a fraction into its simplest/reduced form doesn't change its value, both the original (unsimplified) fraction and its reduced form represent the same exact value/quantity. So, &lt;math&gt;\tfrac{2}{4}&lt;/math&gt; and &lt;math&gt;\tfrac{1}{2}&lt;/math&gt; represents the same quantity, a half! 

Here, we will present a few fractions for you to simplify.
====Sample problems for ''simplifying fractions'' (use ''/'' as the fraction line)====
&lt;quiz display=simple points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{6}{8}=&lt;/math&gt;{ 3/4_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{4}{60}=&lt;/math&gt;{ 1/15_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{30}{90}=&lt;/math&gt;{ 1/3_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{8}{18}=&lt;/math&gt;{ 4/9_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{9}{72}=&lt;/math&gt;{ 1/8_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{64}{46}=&lt;/math&gt;{ 32/23|1 9/23_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{206}{340}=&lt;/math&gt;{ 103/170_7 }
&lt;/quiz&gt;
===== Adding or Subtracting Fractions =====
[[File:Fractionsworkalgebra.PNG|thumb|right|What we just worked on, summarized]]
To simply add or subtract fractions, make sure the denominators of the fractions you are adding or subtracting are the same. If they are not, find the least common denominator (LCD). For example, if you want to add &lt;math&gt;\tfrac{4}{2}&lt;/math&gt; and &lt;math&gt;\tfrac{4}{6}&lt;/math&gt;, you first have to multiply the 2 in &lt;math&gt;\tfrac{4}{2}&lt;/math&gt; by '''3''', which equals '''6'''... BUT you cannot just multiply 2 only, you also have to multiply 4 by 3, since that's what you did to 2, the denominator. If you change the denominator, you have to change the numerator. ('''This step is crucial as it allows you to preserve the same value of the fraction''' but with just a different representation)

Alright, we got that out of the way, so once we have &lt;math&gt;\tfrac{12}{6}&lt;/math&gt; + &lt;math&gt;\tfrac{4}{6}&lt;/math&gt;, we can simply add. So &lt;math&gt;12&lt;/math&gt; + &lt;math&gt;4&lt;/math&gt; = &lt;math&gt;16&lt;/math&gt;, but don't add the denominators, they stay the same. So the answer is &lt;math&gt;\tfrac{16}{6}&lt;/math&gt;, and then we simplify down to &lt;math&gt;\tfrac{8}{3}&lt;/math&gt; dividing by 2 on both the numerator and denominator. 

But... did you notice something? &lt;math&gt;\tfrac{8}{3}&lt;/math&gt;? That doesn't seem right, does it? The denominator is smaller than the numerator. When you have a fraction like this, you have to convert it to a '''mixed fraction''' (skip to [[Speak_Math_Now!/Week_1:_Introduction_To_Algebra#Improper_Fraction_--.3E_Mixed_Fraction|section 2.1.1.4]]).

===== Multiplying Fractions =====
To multiply fractions, its easiest to first simplify your fraction to simplest terms. Once you have done that, you can simply multiply the numerators and the denominators. And obviously, simplify your final product, if you can. So, we have &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; and &lt;math&gt;\tfrac{2}{6}&lt;/math&gt;. You could multiply the numerators and denominators straight away and simplify at the end if you are comfortable, but to make it easier and clearer, we should simplify the fractions first. We simplify 6 and 8 by dividing both by 2, we also divide 2 and 6 by 2. So the fractions are now &lt;math&gt;\tfrac{3}{4}&lt;/math&gt; and &lt;math&gt;\tfrac{1}{3}&lt;/math&gt;. You simply multiply those two fractions by multiplying the numerator by the numerator, and doing the same for the denominators. After completing this process, you will get a solution (in fraction form). &lt;math&gt;\tfrac{3}{4}&lt;/math&gt; × &lt;math&gt;\tfrac{1}{3}&lt;/math&gt; &lt;math&gt;=&lt;/math&gt; &lt;math&gt;\tfrac{3}{12}&lt;/math&gt;. 

&lt;math&gt;\tfrac{3}{12}&lt;/math&gt; is not going to be our final product, though, since we can simplify the fraction by dividing the fraction by 3, which results in &lt;math&gt;\tfrac{1}{4}&lt;/math&gt;.

===== Dividing Fractions =====
There is an interesting twist when it comes to dividing fractions. You have to turn the fraction you want to divide by (second fraction) upside-down, also known as &quot;Keep, Change, Flip&quot; where you keep the first fraction the same, change the operation to multiplication, and replace the second fractions numerator with the denominator and the denominator with the numerator. Not only that, you have to turn the division symbol (÷) into a multiplication symbol (× or •). After that, you use your skills you learned in multiplying a fraction, and you multiply both of the fractions. Simplify if you need to.

So, &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; ÷ &lt;math&gt;\tfrac{7}{12}&lt;/math&gt;. Change the division symbol to a multiplication symbol, and turn the fraction you want to divide by upside-down (the upside-down fraction is known as a '''reciprocal'''). So &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; × (or •) &lt;math&gt;\tfrac{12}{7}&lt;/math&gt;. Multiply the numerators and denominators. The answer is &lt;math&gt;\tfrac{72}{56}&lt;/math&gt;, simplified down to &lt;math&gt;\tfrac{9}{7}&lt;/math&gt;.

===== Improper Fraction --&gt; Mixed Fraction =====
Divide the numerator by the denominator. The '''quotient''' (result of the division taking place/number above the division line) will be the whole number of the mixed fraction, while the numerator will be the remainder. The denominator remains unchanged, so don't change the denominator at all!

{{notice|If you would like to take the quiz on Fractions, please go to '''[[Speak Math Now!/Week 1: Introduction To Algebra/Fractions Quiz]]'''}}

See also: https://www.tes.com/lessons/bJieZ4sFPJbSTw/fractions-4-mixed-numbers-and-improper-fractions

===Decimals===

Ever wondered how to write 8&lt;math&gt;\tfrac{47}{100}&lt;/math&gt; as a decimal? Well, you've got the answer: 8.47! How did we get that answer? Let's look at a few more and maybe you'll see the pattern:
# 6&lt;math&gt;\tfrac{98}{100}&lt;/math&gt; = 6.98
# 2&lt;math&gt;\tfrac{56}{100}&lt;/math&gt; = 2.56
# 9&lt;math&gt;\tfrac{27}{100}&lt;/math&gt; = 9.27
# 5&lt;math&gt;\tfrac{83}{100}&lt;/math&gt; = 5.83
You see? We simply put the mixed number in front of the dot, and with the numerator, we slap that behind the dot! Throw out the 100, it's not important when building your decimal. 

Decimals are all about place value, the value of a number in a specific place in a number. So, when we have &lt;math&gt;6.72&lt;/math&gt;, the &lt;math&gt;6&lt;/math&gt; is in the Ones place. Now, let's throw &lt;math&gt;9&lt;/math&gt; in the tens place, which is 10 times bigger than the Ones place: &lt;math&gt;96.72&lt;/math&gt;. But... that's doesn't seem enough, does it? Let's throw in a &lt;math&gt;6, 2, 8&lt;/math&gt; and a &lt;math&gt;3&lt;/math&gt; in there! And now, we have: &lt;math&gt;628,396.72&lt;/math&gt;.

Woah! That's a pretty big number, but we can easily break this number down to it's place value. Let's do it!

So, our number, &lt;math&gt;628,396.72&lt;/math&gt;, is the number we need to break down. Let's start from the decimal point, and move left:
* The number &lt;math&gt;6&lt;/math&gt; is in the Ones place. '''x10'''
* The number &lt;math&gt;9&lt;/math&gt; is in the Tens place. '''x10'''
* The number &lt;math&gt;3&lt;/math&gt; is in the Hundreds place. '''x10'''
* The number &lt;math&gt;8&lt;/math&gt; is in the Thousands place. '''x10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Ten thousands place. '''x10'''
* The number &lt;math&gt;6&lt;/math&gt; is in the Hundred Thousands place.

Now we have broken up the numbers left of the decimal--What about the numbers on the ''right''? Let's throw in a &lt;math&gt;5, 2, 4&lt;/math&gt; and a &lt;math&gt;7&lt;/math&gt;. Now, we have &lt;math&gt;628,396.725,247&lt;/math&gt;. Let's break this number up like we did above.

So, our number, &lt;math&gt;628,396.725,247&lt;/math&gt;, is the number we need to break down. This time, we need to start on the decimal point, and move ''right'':
* The number &lt;math&gt;7&lt;/math&gt; is in the Tenths place. '''x-10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Hundredths place. '''x-10'''
* The number &lt;math&gt;5&lt;/math&gt; is in the Thousandths place. '''x-10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Ten Thousandths place. '''x-10'''
* The number &lt;math&gt;4&lt;/math&gt; is in the Hundred Thousandths place. '''x-10'''
* The number &lt;math&gt;7&lt;/math&gt; is in the Millionths place. 

We have just now gone over the importance of Place Value in the Decimal World. Now, we will go into how to work with decimals, in the Decimal World!

See also: http://www.shmoop.com/fractions-decimals/place-value-naming-decimals.html

==== Adding/Subtracting Decimals ====
To add decimals, in addition column-style, put the decimals in its place with the decimals lined up. Then simply add on. So, for &lt;math&gt;1.5&lt;/math&gt; + &lt;math&gt;2.5&lt;/math&gt; we'd line up the decimal points. But, if we had a problem like &lt;math&gt;1.15&lt;/math&gt; + &lt;math&gt;2.0&lt;/math&gt;, we'd add a &lt;math&gt;0&lt;/math&gt; after the &lt;math&gt;0&lt;/math&gt; that is behind the decimal. Adding a zero to a place in a decimal means &quot;no value&quot;. So &lt;math&gt;10&lt;/math&gt; basically means no ones, and &lt;math&gt;100&lt;/math&gt;, means no ones or hundreds. Same things goes for subtracting as well folks.
=====Sample problems for ''adding/subtracting decimals''=====
&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
6.8 - 2.5 = { 4.3_6 }

{
|type=&quot;{}&quot;}
3.4 + 5.6 = { 9_6 }

{
|type=&quot;{}&quot;}
9 + 4.50 = { 13.5_6 }

{
|type=&quot;{}&quot;}
41.89 + 25.00 = { 66.89_6 }

{
|type=&quot;{}&quot;}
9.01 + 3.089 = { 12.099_6 }

{
|type=&quot;{}&quot;}
10.90 + 11.1 = { 22_6 }

{
|type=&quot;{}&quot;}
9.5 + 3.44 = { 12.94_6 }

{
|type=&quot;{}&quot; coef=&quot;2.5&quot;}
9.00 x 2.00 = { 18_6 } &lt;big&gt;(BONUS!)&lt;/big&gt; 
&lt;/quiz&gt;

==== Multiplying Decimals ====
[[File:9.82x5.73 multiplication image.svg|thumb|A visual representation of the multiplication example]]
Multiplying decimals isn't as hard as it really seems to be. So, we have &lt;math&gt;9.83&lt;/math&gt; × &lt;math&gt;5.73&lt;/math&gt;. For most people, column multiplication is a lot easier than side-by-side multiplication. That being mentioned, let us column these numbers:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;× &lt;math&gt;5.73&lt;/math&gt;
-------

Now that we have our problem, we should simply ignore the decimal points and just multiply as usual, so you should get this answer once you are done with that (remember to add a zero (and grow with zeros in each line) to each and every line of addition):

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;        × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt; &lt;br&gt;  &lt;math&gt;+&lt;/math&gt;   &lt;math&gt;68810&lt;/math&gt; &lt;br&gt;  &lt;math&gt;+&lt;/math&gt; &lt;math&gt;491500
&lt;/math&gt;
-------

With the simple usage of addition, we should get:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;        × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt;&lt;br&gt;   &lt;math&gt;+&lt;/math&gt;  &lt;math&gt;68810&lt;/math&gt;&lt;br&gt;  &lt;math&gt;+
&lt;/math&gt; &lt;math&gt;491500&lt;/math&gt;
-------
&lt;math&gt;563259&lt;/math&gt;

Now, we need to bring back our handy dandy decimal point, but where? In &lt;math&gt;9.83&lt;/math&gt; and &lt;math&gt;5.73&lt;/math&gt;, there are FOUR numbers in these 2 numbers overall that are behind the decimal point (in each number, there are two numbers behind the decimal points). So, we have &lt;math&gt;9.83&lt;/math&gt; and &lt;math&gt;5.73&lt;/math&gt;. Now, that totals up to four numbers overall behind the decimal point. So in &lt;math&gt;563259&lt;/math&gt;, we need to move the decimal point four times (beginning from the right). So watch as follows:

&lt;math&gt;563259.&lt;/math&gt;&lt;br&gt;
&lt;math&gt;56325.9&lt;/math&gt;&lt;br&gt;
&lt;math&gt;5632.59&lt;/math&gt;&lt;br&gt;
&lt;math&gt;563.259&lt;/math&gt;&lt;br&gt;
&lt;math&gt;56.3259&lt;/math&gt;

That simple. Now, review your work, your whole work should look like this:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;       × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt;&lt;br&gt;    &lt;math&gt;+&lt;/math&gt;&lt;math&gt;68810&lt;/math&gt;&lt;br&gt;  &lt;math&gt;+&lt;/math&gt;&lt;math&gt;491500&lt;/math&gt;
-------
&lt;math&gt;56.3259&lt;/math&gt;

==== Dividing Decimals ====
;Dividing a decimal by a whole number
If you want to divide a decimal by a whole number, you should divide the 2 numbers, omitting the decimal point. After you are done dividing, add the decimal point to the '''quotient''' (final product/answer at the top of the long division symbol). The decimal should be right above the decimal point in the '''dividend''' (number in the box/number that is being divided). It's quite easy and simple, as long as you know how to do long division and if you are still familiar with long division. 

Hey, this seems ''too'' easy--Let's figure out how to divide a decimal by a decimal!
;Dividing a decimal by a decimal
The trick to dividing a decimal by a decimal is to shift the decimal point as many times as it gets to a whole number, so follow along: &lt;math&gt;69.45&lt;/math&gt; ÷ &lt;math&gt;5.78&lt;/math&gt;. Now, we simply move the decimal point as many times as we need to make the number we are going to use to divide 69.45 a whole number, so watch as followed:&lt;br&gt;
&lt;math&gt;69.45&lt;/math&gt; ÷ &lt;math&gt;5.78&lt;/math&gt; →&lt;br&gt;
&lt;math&gt;694.5&lt;/math&gt; ÷ &lt;math&gt;57.8&lt;/math&gt; →&lt;br&gt;
&lt;math&gt;6945&lt;/math&gt;. ÷ &lt;math&gt;578&lt;/math&gt;.


Now that we have finally got our dividend a whole number (and now our first number that we are going to divide), we can go ahead and divide normally (using long division). In the end, &lt;math&gt;69.45&lt;/math&gt; divided by &lt;math&gt;5.78&lt;/math&gt; should get you &lt;math&gt;12.0155709&lt;/math&gt;!

A pretty simple one we could go is &lt;math&gt;6.4&lt;/math&gt; ÷ &lt;math&gt;0.4&lt;/math&gt;, here, we simply move our dots like so:&lt;br&gt;
&lt;math&gt;6.4&lt;/math&gt; ÷ &lt;math&gt;0.4&lt;/math&gt;&lt;br&gt;
&lt;math&gt;64&lt;/math&gt; ÷ &lt;math&gt;04.&lt;/math&gt;&lt;br&gt;
&lt;math&gt;64&lt;/math&gt; ÷ &lt;math&gt;4&lt;/math&gt;&lt;br&gt;
Then, we can simply divide, heck... we don't even need to do long division! The answer should pop in your head, which is &lt;math&gt;16&lt;/math&gt;.

{{notice|If you would like to take the quiz on Decimals, please go to '''[[Speak Math Now!/Week 1: Introduction To Algebra/Decimals Quiz]]'''}}

===Percentages===
A good definition of &quot;percent&quot; is a fraction in which the denominator is the number &lt;math&gt;100&lt;/math&gt;. For example, the numbers &lt;math&gt;59%&lt;/math&gt;, &lt;math&gt;63%&lt;/math&gt;, &lt;math&gt;91%&lt;/math&gt;, and &lt;math&gt;85%&lt;/math&gt;, are the same as just saying &lt;math&gt;\tfrac{59}{100}&lt;/math&gt;, &lt;math&gt;\tfrac{63}{100}&lt;/math&gt;, &lt;math&gt;\tfrac{91}{100}&lt;/math&gt;, and &lt;math&gt;\tfrac{85}{100}&lt;/math&gt;. You could also say 59 out of 100 parts, 63 out of 100 parts, 91 out of 100 parts, and 85 out of 100 parts.
====Converting Percentages====
Now that we got the basis of percentages and how they operate, we should look into changing percentages.
===== Percentage → Decimal =====
Let's look in turning a percentage into a decimal point first. It's very simple. Let's say you have &lt;math&gt;\tfrac{9}{100}&lt;/math&gt;, which, in percentage form, is &lt;math&gt;9%&lt;/math&gt;. So, we have 9%. Now, we want to change it to a decimal (I don't know, think of a reason). We simply convert the percentage symbol into a decimal point, so like this: &lt;math&gt;9.&lt;/math&gt;. Now, we have &lt;math&gt;9.&lt;/math&gt;, so then we move the decimal number two places to the left, like so: &lt;math&gt;9.&lt;/math&gt; → &lt;math&gt;.9&lt;/math&gt; → &lt;math&gt;.09&lt;/math&gt;. So now, we have &lt;math&gt;0.09&lt;/math&gt;. We added the 2 zeros in because there is no value in the tenths place, and because &lt;math&gt;.09&lt;/math&gt; does not look quite right. Looks a bit off.
===== Samples problems for ''converting percentages to decimals'' =====
&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
59% = { 0.59_5 }

{
|type=&quot;{}&quot;}
63% = { 0.63_5 }

{
|type=&quot;{}&quot;}
91% = { 0.91_5 }

{
|type=&quot;{}&quot;}
85% = { 0.85_5 }

{
|type=&quot;{}&quot;}
9% = { 0.09_5 }

{
|type=&quot;{}&quot;}
9834% = { 98.34_5 }

{
|type=&quot;{}&quot;}
20% = { 0.2_5 }

{
|type=&quot;{}&quot;}
4% = { 0.04_5 }

{
|type=&quot;{}&quot;}
7.6% = { 0.076_5 }

{
|type=&quot;{}&quot;}
6% = { 0.06_5 }
&lt;/quiz&gt;
===== Decimal → Percentage =====
Now to convert a decimal into percentage we essentially do the complete opposite. We have &lt;math&gt;98.34&lt;/math&gt;. We need this to be a percentage (easier to read). Move the decimal point two places to the right. So, watch: &lt;math&gt;98.34&lt;/math&gt; → &lt;math&gt;983.4&lt;/math&gt; → &lt;math&gt;9834.&lt;/math&gt; --Now, we have &lt;math&gt;9834.&lt;/math&gt;, but the decimal point, since it's now a percentage, should not be there, but instead, a percentage should talk the decimal point's place. Now, we have our final result of &lt;math&gt;9834%&lt;/math&gt;.

==== Finding percent of a number ====
[[File:Universität Bonn.jpg|thumb|right|Would this be the fictional university these students were trying to get accepted to?]]
So, 95 students applied to a university (the fictional [[User:Atcovi/Mustafa Einhoonansebadoi University|Mustafa Einhoonansebadoi University]], for example), and only 20% of the students made it. 20%? What? With this in mind, we want to find &lt;math&gt;20%&lt;/math&gt; of &lt;math&gt;95&lt;/math&gt;. We take the percentage, &lt;math&gt;20%&lt;/math&gt;, and divide it by &lt;math&gt;100&lt;/math&gt;. So we get &lt;math&gt;20/100&lt;/math&gt; = &lt;math&gt;.2&lt;/math&gt;. Then, we multiply &lt;math&gt;.2&lt;/math&gt; by &lt;math&gt;95&lt;/math&gt;, in which we get &lt;math&gt;19&lt;/math&gt;. So &lt;math&gt;20%&lt;/math&gt; of &lt;math&gt;95&lt;/math&gt; is &lt;math&gt;19&lt;/math&gt;. Therefore, only 19 students out of 95 students made it into the fictional Mustafa Einhoonansebadoi University. 

{{subpage navbar}}
[[Category:Speak Math Now!]]</text>
      <sha1>0frw34lye6k0xlgz0135kjc8dzjeac5</sha1>
    </revision>
    <revision>
      <id>2807865</id>
      <parentid>2807863</parentid>
      <timestamp>2026-05-07T09:28:45Z</timestamp>
      <contributor>
        <username>~2026-27598-14</username>
        <id>3070872</id>
      </contributor>
      <origin>2807865</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21774" sha1="fxyyo9wepgw08nuud6j6it29skyt4ru" xml:space="preserve">[[File:Quadratic formula.svg|thumb|right|A example of a Algebra formula (quadratic formula)]]
{{mathematics}}
{{secondary education}}
{{lesson}}
{{complete}}

'''Algebra''' (from the Arabic word &quot;al-jabr&quot; (الجبر), meaning &quot;reunion of broken parts&quot;) can feel like quite a complicated language of mathematics. However, as time goes on, completing Algebra will get easier and easier until it's a breeze. Completing Algebra takes true dedication with a worthwhile reward. This week, we will get into what Algebra is, and some warm ups (on arithmetic). Even though this may seem pointless, it is IMPORTANT that you review through these warm ups and get comfortable in solving them to lay a strong foundation for understanding larger topics later on.

Without further do, let's dig right into this!
==Algebra==
===What is Algebra?===
[[File:AlgebraJournalWork11-14-16.jpg|thumb|left|You might have to do this much work for a small answer!]]
[[File:WhatwejustdidinthisproblemALGEBRA.PNG|thumb|right|This, summed up algebraically, is what we just did]]
A broad part of mathematics. In summary, Algebra solves for values that are not known yet. Such as □ - 4 = 6... instead of this empty box, we replace that with something known as a '''variable''', which is a letter that is used for something we do not know yet (of it's value). 

So, for the problem: □ - 4 = 6... we will have to do some math! We need to add positive 4, to the negative 4 on the left handside of the equation. When you add a number to another number (in which the solution becomes 0), you have cancelled out that number. But, you cannot just cancel out 4, and be done with it! You have to also add 4 to the number 6 on the righthand side of the equal sign, which brings out a golden rule in Algebra: '''To keep the balance, what we do to one side of the &quot;=&quot; we should also do to the other side!&quot;. ''

Now... once you have added 6 + 4 (which is 10), you should get □ = 10.

Great! You've done what's called &quot;solving for x&quot;, but in this case, it's an empty box... well guys, you're going to have to throw out that box, because it will be represented by a variable (a letter, from a - z). The letter that is commonly used for variables is x (and the reason for this  dates back to the origin of Algebra itself; Muhammad ibn Musa al-Khwarizmi, one of the main &quot;founders&quot; of Algebra you could say, used to call the unknown or &quot;box&quot; in our example, &quot;'''shay'''&quot;. &quot;'''Shay&quot;''' comes from the Arabic word '''شَيْء''', which essentially means &quot;thing&quot;. When Al-Khwarizmi's works were translated to Latin in medieval Spain, &quot;shay&quot; was translated as '''&quot;xay&quot;,''' since the letter x was pronounced as sh in Spain. Later on, this word &quot;'''xay&quot;''' got abbreviated to &quot;'''x&quot;''' to repesent the symbol of the unkown. Source/For more information, check out this [https://www.pbs.org/empires/islam/innoalgebra.html PBS] page). Brief history lesson aside, it's interesting to learn why &quot;x&quot; is the default norm when solving for the unkown in Algebra! '''Importantly, know that the unknown can take on any letter or variable name, not just x (i.e. could be a, b, c, y, even words if you wanted, etc.). Just remember what that variable is representing (the unknown value in this case).''' 

So instead of □ = 10, it's actually '''x = 10'''.

===== Sample problems of ''solving for x'' =====
&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{''x'' − 9 = 20
|type=&quot;{}&quot;}
''x''={ 29_3 }

{''x'' − 3 = 6
|type=&quot;{}&quot;}
''x''={ 9_3 }

{''x'' + 5 = 15
|type=&quot;{}&quot;}
''x''={ 10_3 }

{''x'' + 17 = 23
|type=&quot;{}&quot;}
''x''={ 6_3 }

{4''x'' = 12
|type=&quot;{}&quot;}
''x''={ 3_3 }

{''x''/2 = 0.5
|type=&quot;{}&quot;}
''x''={ 1_3 }

{''x''/50 = 2
|type=&quot;{}&quot;}
''x''={ 100_3 }

{''x''/9 = 5
|type=&quot;{}&quot;}
''x''={ 45_3 }
&lt;/quiz&gt;

Seems simple, huh? Well, it will get complicated, which is why it is important for you to do some review of your arithmetic! Let's dig into that...

==Arithmetic==
[[File:Multiply 4 bags 3 marbles.svg|thumb|right|4 x 3 = 12 (multiplication)]]
'''Arithmetic''' has to deal with elementary/basic levels of math, such as division, multiplication, subtraction, and addition. Basically, just working with numbers.

This SHOULD be a level familiar with you. If you are not familiar with arithmetic math/rules, then PLEASE review through Arithmetic, as you won't survive even the 1st step of Algebra. Trust me, the basics are THAT important.
===Fractions===
[[File:Cake quarters.svg|thumb|left|A Cake with fractions]]
'''Fractions''' (from Latin ''fractus'', &quot;broken&quot;) are parts of a whole. On the left side in the image of the cake, there is only &lt;math&gt;3/4&lt;/math&gt;'s of the cake showing, the other &lt;math&gt;1/4&lt;/math&gt; has been eaten/taken away.

The number, ''3'', in &lt;math&gt;3/4&lt;/math&gt;, is what is known as a '''numerator''' (Numerator: Number at the top, tells us of how much of the number is being talked about/being used). The number, ''4'', in &lt;math&gt;3/4&lt;/math&gt;, is what is known as a '''denominator''' (Denominator: Number showing the all time total).
;
;Simplest form/reduced form
A reduced form of a fraction is a fraction that cannot be divided by any number other than 1, and the denominator is greater than 1. So &lt;math&gt;2/4&lt;/math&gt; is NOT in simplest form, since we can divide 2 and 4, by 2... which results in the following number: &lt;math&gt;1/2&lt;/math&gt;. Though, not every fraction can be divided by 2, there are fractions, such as: &lt;math&gt;5/35&lt;/math&gt;, &lt;math&gt;7/21&lt;/math&gt;, and &lt;math&gt;30/5&lt;/math&gt;. The two first fractions are not divisible by 2, and &lt;math&gt;30/5&lt;/math&gt; can not be divided by 2 on both sides, but only on &lt;math&gt;30&lt;/math&gt;. It's important to simplify as if you were in a test, your teacher will mark your problems as incorrect if you didn't simplify your fractions. Keep in mind that simplifying a fraction into its simplest/reduced form doesn't change its value, both the original (unsimplified) fraction and its reduced form represent the same exact value/quantity. So, &lt;math&gt;\tfrac{2}{4}&lt;/math&gt; and &lt;math&gt;\tfrac{1}{2}&lt;/math&gt; represents the same quantity, a half! 

Here, we will present a few fractions for you to simplify.
====Sample problems for ''simplifying fractions'' (use ''/'' as the fraction line)====
&lt;quiz display=simple points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{6}{8}=&lt;/math&gt;{ 3/4_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{4}{60}=&lt;/math&gt;{ 1/15_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{30}{90}=&lt;/math&gt;{ 1/3_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{8}{18}=&lt;/math&gt;{ 4/9_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{9}{72}=&lt;/math&gt;{ 1/8_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{64}{46}=&lt;/math&gt;{ 32/23|1 9/23_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{206}{340}=&lt;/math&gt;{ 103/170_7 }
&lt;/quiz&gt;
===== Adding or Subtracting Fractions =====
[[File:Fractionsworkalgebra.PNG|thumb|right|What we just worked on, summarized]]
To simply add or subtract fractions, make sure the denominators of the fractions you are adding or subtracting are the same. If they are not, find the least common denominator (LCD). For example, if you want to add &lt;math&gt;\tfrac{4}{2}&lt;/math&gt; and &lt;math&gt;\tfrac{4}{6}&lt;/math&gt;, you first have to multiply the 2 in &lt;math&gt;\tfrac{4}{2}&lt;/math&gt; by '''3''', which equals '''6'''... BUT you cannot just multiply 2 only, you also have to multiply 4 by 3, since that's what you did to 2, the denominator. If you change the denominator, you have to change the numerator. ('''This step is crucial as it allows you to preserve the same value of the fraction''' but with just a different representation)

Alright, we got that out of the way, so once we have &lt;math&gt;\tfrac{12}{6}&lt;/math&gt; + &lt;math&gt;\tfrac{4}{6}&lt;/math&gt;, we can simply add. So &lt;math&gt;12&lt;/math&gt; + &lt;math&gt;4&lt;/math&gt; = &lt;math&gt;16&lt;/math&gt;, but don't add the denominators, they stay the same. So the answer is &lt;math&gt;\tfrac{16}{6}&lt;/math&gt;, and then we simplify down to &lt;math&gt;\tfrac{8}{3}&lt;/math&gt; dividing by 2 on both the numerator and denominator. 

But... did you notice something? &lt;math&gt;\tfrac{8}{3}&lt;/math&gt;? That doesn't seem right, does it? The denominator is smaller than the numerator. When you have a fraction like this, you have to convert it to a '''mixed fraction''' (skip to [[Speak_Math_Now!/Week_1:_Introduction_To_Algebra#Improper_Fraction_--.3E_Mixed_Fraction|section 2.1.1.4]]).

===== Multiplying Fractions =====
To multiply fractions, its easiest to first simplify your fraction to simplest terms. Once you have done that, you can simply multiply the numerators and the denominators. And obviously, simplify your final product, if you can. So, we have &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; and &lt;math&gt;\tfrac{2}{6}&lt;/math&gt;. You could multiply the numerators and denominators straight away and simplify at the end if you are comfortable, but to make it easier and clearer, we should simplify the fractions first. We simplify 6 and 8 by dividing both by 2, we also divide 2 and 6 by 2. So the fractions are now &lt;math&gt;\tfrac{3}{4}&lt;/math&gt; and &lt;math&gt;\tfrac{1}{3}&lt;/math&gt;. You simply multiply those two fractions by multiplying the numerator by the numerator, and doing the same for the denominators. After completing this process, you will get a solution (in fraction form). &lt;math&gt;\tfrac{3}{4}&lt;/math&gt; × &lt;math&gt;\tfrac{1}{3}&lt;/math&gt; &lt;math&gt;=&lt;/math&gt; &lt;math&gt;\tfrac{3}{12}&lt;/math&gt;. 

&lt;math&gt;\tfrac{3}{12}&lt;/math&gt; is not going to be our final product, though, since we can simplify the fraction by dividing the fraction by 3, which results in &lt;math&gt;\tfrac{1}{4}&lt;/math&gt;.

===== Dividing Fractions =====
There is an interesting twist when it comes to dividing fractions. You have to turn the fraction you want to divide by (second fraction) upside-down, also known as &quot;Keep, Change, Flip&quot; where you keep the first fraction the same, change the operation to multiplication, and replace the second fractions numerator with the denominator and the denominator with the numerator. Not only that, you have to turn the division symbol (÷) into a multiplication symbol (× or •). After that, you use your skills you learned in multiplying a fraction, and you multiply both of the fractions. Simplify if you need to.

So, &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; ÷ &lt;math&gt;\tfrac{7}{12}&lt;/math&gt;. Change the division symbol to a multiplication symbol, and turn the fraction you want to divide by upside-down (the upside-down fraction is known as a '''reciprocal'''). So &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; × (or •) &lt;math&gt;\tfrac{12}{7}&lt;/math&gt;. Multiply the numerators and denominators. The answer is &lt;math&gt;\tfrac{72}{56}&lt;/math&gt;, simplified down to &lt;math&gt;\tfrac{9}{7}&lt;/math&gt;.

===== Improper Fraction --&gt; Mixed Fraction =====
Divide the numerator by the denominator. The '''quotient''' (result of the division taking place/number above the division line) will be the whole number of the mixed fraction, while the numerator will be the remainder. The denominator remains unchanged, so don't change the denominator at all!

{{notice|If you would like to take the quiz on Fractions, please go to '''[[Speak Math Now!/Week 1: Introduction To Algebra/Fractions Quiz]]'''}}

See also: https://www.tes.com/lessons/bJieZ4sFPJbSTw/fractions-4-mixed-numbers-and-improper-fractions

===Decimals===

Ever wondered how to write 8&lt;math&gt;\tfrac{47}{100}&lt;/math&gt; as a decimal? Well, you've got the answer: 8.47! How did we get that answer? Let's look at a few more and maybe you'll see the pattern:
# 6&lt;math&gt;\tfrac{98}{100}&lt;/math&gt; = 6.98
# 2&lt;math&gt;\tfrac{56}{100}&lt;/math&gt; = 2.56
# 9&lt;math&gt;\tfrac{27}{100}&lt;/math&gt; = 9.27
# 5&lt;math&gt;\tfrac{83}{100}&lt;/math&gt; = 5.83
You see? We simply put the mixed number in front of the dot, and with the numerator, we slap that behind the dot! Throw out the 100, it's not important when building your decimal. 

Decimals are all about place value, the value of a number in a specific place in a number. So, when we have &lt;math&gt;6.72&lt;/math&gt;, the &lt;math&gt;6&lt;/math&gt; is in the Ones place. Now, let's throw &lt;math&gt;9&lt;/math&gt; in the tens place, which is 10 times bigger than the Ones place: &lt;math&gt;96.72&lt;/math&gt;. But... that's doesn't seem enough, does it? Let's throw in a &lt;math&gt;6, 2, 8&lt;/math&gt; and a &lt;math&gt;3&lt;/math&gt; in there! And now, we have: &lt;math&gt;628,396.72&lt;/math&gt;.

Woah! That's a pretty big number, but we can easily break this number down to it's place value. Let's do it!

So, our number, &lt;math&gt;628,396.72&lt;/math&gt;, is the number we need to break down. Let's start from the decimal point, and move left:
* The number &lt;math&gt;6&lt;/math&gt; is in the Ones place. '''x10'''
* The number &lt;math&gt;9&lt;/math&gt; is in the Tens place. '''x10'''
* The number &lt;math&gt;3&lt;/math&gt; is in the Hundreds place. '''x10'''
* The number &lt;math&gt;8&lt;/math&gt; is in the Thousands place. '''x10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Ten thousands place. '''x10'''
* The number &lt;math&gt;6&lt;/math&gt; is in the Hundred Thousands place.

Now we have broken up the numbers left of the decimal--What about the numbers on the ''right''? Let's throw in a &lt;math&gt;5, 2, 4&lt;/math&gt; and a &lt;math&gt;7&lt;/math&gt;. Now, we have &lt;math&gt;628,396.725,247&lt;/math&gt;. Let's break this number up like we did above.

So, our number, &lt;math&gt;628,396.725,247&lt;/math&gt;, is the number we need to break down. This time, we need to start on the decimal point, and move ''right'':
* The number &lt;math&gt;7&lt;/math&gt; is in the Tenths place. '''x-10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Hundredths place. '''x-10'''
* The number &lt;math&gt;5&lt;/math&gt; is in the Thousandths place. '''x-10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Ten Thousandths place. '''x-10'''
* The number &lt;math&gt;4&lt;/math&gt; is in the Hundred Thousandths place. '''x-10'''
* The number &lt;math&gt;7&lt;/math&gt; is in the Millionths place. 

We have just now gone over the importance of Place Value in the Decimal World. Now, we will go into how to work with decimals, in the Decimal World!

See also: http://www.shmoop.com/fractions-decimals/place-value-naming-decimals.html

==== Adding/Subtracting Decimals ====
To add decimals, in addition column-style, put the decimals in its place with the decimals lined up. Then simply add on. So, for &lt;math&gt;1.5&lt;/math&gt; + &lt;math&gt;2.5&lt;/math&gt; we'd line up the decimal points. But, if we had a problem like &lt;math&gt;1.15&lt;/math&gt; + &lt;math&gt;2.0&lt;/math&gt;, we'd add a &lt;math&gt;0&lt;/math&gt; after the &lt;math&gt;0&lt;/math&gt; that is behind the decimal. Adding a zero to a place in a decimal means &quot;no value&quot;. So &lt;math&gt;10&lt;/math&gt; basically means no ones, and &lt;math&gt;100&lt;/math&gt;, means no ones or hundreds. Same things goes for subtracting as well folks.
=====Sample problems for ''adding/subtracting decimals''=====
&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
6.8 - 2.5 = { 4.3_6 }

{
|type=&quot;{}&quot;}
3.4 + 5.6 = { 9_6 }

{
|type=&quot;{}&quot;}
9 + 4.50 = { 13.5_6 }

{
|type=&quot;{}&quot;}
41.89 + 25.00 = { 66.89_6 }

{
|type=&quot;{}&quot;}
9.01 + 3.089 = { 12.099_6 }

{
|type=&quot;{}&quot;}
10.90 + 11.1 = { 22_6 }

{
|type=&quot;{}&quot;}
9.5 + 3.44 = { 12.94_6 }

{
|type=&quot;{}&quot; coef=&quot;2.5&quot;}
9.00 x 2.00 = { 18_6 } &lt;big&gt;(BONUS!)&lt;/big&gt; 
&lt;/quiz&gt;

==== Multiplying Decimals ====
[[File:9.82x5.73 multiplication image.svg|thumb|A visual representation of the multiplication example]]
Multiplying decimals isn't as hard as it really seems to be. So, we have &lt;math&gt;9.83&lt;/math&gt; × &lt;math&gt;5.73&lt;/math&gt;. For most people, column multiplication is a lot easier than side-by-side multiplication. That being mentioned, let us column these numbers:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;× &lt;math&gt;5.73&lt;/math&gt;
-------

Now that we have our problem, we should simply ignore the decimal points and just multiply as usual, so you should get this answer once you are done with that (remember to add a zero (and grow with zeros in each line) to each and every line of addition):

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;        × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt; &lt;br&gt;  &lt;math&gt;+&lt;/math&gt;   &lt;math&gt;68810&lt;/math&gt; &lt;br&gt;  &lt;math&gt;+&lt;/math&gt; &lt;math&gt;491500
&lt;/math&gt;
-------

With the simple usage of addition, we should get:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;        × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt;&lt;br&gt;   &lt;math&gt;+&lt;/math&gt;  &lt;math&gt;68810&lt;/math&gt;&lt;br&gt;  &lt;math&gt;+
&lt;/math&gt; &lt;math&gt;491500&lt;/math&gt;
-------
&lt;math&gt;563259&lt;/math&gt;

Now, we need to bring back our handy dandy decimal point, but where? In &lt;math&gt;9.83&lt;/math&gt; and &lt;math&gt;5.73&lt;/math&gt;, there are FOUR numbers in these 2 numbers overall that are behind the decimal point (in each number, there are two numbers behind the decimal points). So, we have &lt;math&gt;9.83&lt;/math&gt; and &lt;math&gt;5.73&lt;/math&gt;. Now, that totals up to four numbers overall behind the decimal point. So in &lt;math&gt;563259&lt;/math&gt;, we need to move the decimal point four times (beginning from the right). So watch as follows:

&lt;math&gt;563259.&lt;/math&gt;&lt;br&gt;
&lt;math&gt;56325.9&lt;/math&gt;&lt;br&gt;
&lt;math&gt;5632.59&lt;/math&gt;&lt;br&gt;
&lt;math&gt;563.259&lt;/math&gt;&lt;br&gt;
&lt;math&gt;56.3259&lt;/math&gt;

That simple. Now, review your work, your whole work should look like this:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;       × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt;&lt;br&gt;    &lt;math&gt;+&lt;/math&gt;&lt;math&gt;68810&lt;/math&gt;&lt;br&gt;  &lt;math&gt;+&lt;/math&gt;&lt;math&gt;491500&lt;/math&gt;
-------
&lt;math&gt;56.3259&lt;/math&gt;

==== Dividing Decimals ====
;Dividing a decimal by a whole number
If you want to divide a decimal by a whole number, you should divide the 2 numbers, omitting the decimal point. After you are done dividing, add the decimal point to the '''quotient''' (final product/answer at the top of the long division symbol). The decimal should be right above the decimal point in the '''dividend''' (number in the box/number that is being divided). It's quite easy and simple, as long as you know how to do long division and if you are still familiar with long division. 

Hey, this seems ''too'' easy--Let's figure out how to divide a decimal by a decimal!
;Dividing a decimal by a decimal
The trick to dividing a decimal by a decimal is to shift the decimal point as many times as it gets to a whole number, so follow along: &lt;math&gt;69.45&lt;/math&gt; ÷ &lt;math&gt;5.78&lt;/math&gt;. Now, we simply move the decimal point as many times as we need to make the number we are going to use to divide 69.45 a whole number, so watch as followed:&lt;br&gt;
&lt;math&gt;69.45&lt;/math&gt; ÷ &lt;math&gt;5.78&lt;/math&gt; →&lt;br&gt;
&lt;math&gt;694.5&lt;/math&gt; ÷ &lt;math&gt;57.8&lt;/math&gt; →&lt;br&gt;
&lt;math&gt;6945&lt;/math&gt;. ÷ &lt;math&gt;578&lt;/math&gt;.


Now that we have finally got our dividend a whole number (and now our first number that we are going to divide), we can go ahead and divide normally (using long division). In the end, &lt;math&gt;69.45&lt;/math&gt; divided by &lt;math&gt;5.78&lt;/math&gt; should get you &lt;math&gt;12.0155709&lt;/math&gt;!

A pretty simple one we could go is &lt;math&gt;6.4&lt;/math&gt; ÷ &lt;math&gt;0.4&lt;/math&gt;, here, we simply move our dots like so:&lt;br&gt;
&lt;math&gt;6.4&lt;/math&gt; ÷ &lt;math&gt;0.4&lt;/math&gt;&lt;br&gt;
&lt;math&gt;64&lt;/math&gt; ÷ &lt;math&gt;04.&lt;/math&gt;&lt;br&gt;
&lt;math&gt;64&lt;/math&gt; ÷ &lt;math&gt;4&lt;/math&gt;&lt;br&gt;
Then, we can simply divide, heck... we don't even need to do long division! The answer should pop in your head, which is &lt;math&gt;16&lt;/math&gt;.

{{notice|If you would like to take the quiz on Decimals, please go to '''[[Speak Math Now!/Week 1: Introduction To Algebra/Decimals Quiz]]'''}}

===Percentages===
A good definition of &quot;percent&quot; is a fraction in which the denominator is the number &lt;math&gt;100&lt;/math&gt;. For example, the numbers &lt;math&gt;59%&lt;/math&gt;, &lt;math&gt;63%&lt;/math&gt;, &lt;math&gt;91%&lt;/math&gt;, and &lt;math&gt;85%&lt;/math&gt;, are the same as just saying &lt;math&gt;\tfrac{59}{100}&lt;/math&gt;, &lt;math&gt;\tfrac{63}{100}&lt;/math&gt;, &lt;math&gt;\tfrac{91}{100}&lt;/math&gt;, and &lt;math&gt;\tfrac{85}{100}&lt;/math&gt;. You could also say 59 out of 100 parts, 63 out of 100 parts, 91 out of 100 parts, and 85 out of 100 parts.
====Converting Percentages====
Now that we got the basis of percentages and how they operate, we should look into changing percentages.
===== Percentage → Decimal =====
Let's look in turning a percentage into a decimal point first. It's very simple. Let's say you have &lt;math&gt;\tfrac{9}{100}&lt;/math&gt;, which, in percentage form, is &lt;math&gt;9%&lt;/math&gt;. So, we have 9%. Now, we want to change it to a decimal (I don't know, think of a reason). We simply convert the percentage symbol into a decimal point, so like this: &lt;math&gt;9.&lt;/math&gt;. Now, we have &lt;math&gt;9.&lt;/math&gt;, so then we move the decimal number two places to the left, like so: &lt;math&gt;9.&lt;/math&gt; → &lt;math&gt;.9&lt;/math&gt; → &lt;math&gt;.09&lt;/math&gt;. So now, we have &lt;math&gt;0.09&lt;/math&gt;. We added the 2 zeros in because there is no value in the tenths place, and because &lt;math&gt;.09&lt;/math&gt; does not look quite right. Looks a bit off.
===== Samples problems for ''converting percentages to decimals'' =====
&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
59% = { 0.59_5 }

{
|type=&quot;{}&quot;}
63% = { 0.63_5 }

{
|type=&quot;{}&quot;}
91% = { 0.91_5 }

{
|type=&quot;{}&quot;}
85% = { 0.85_5 }

{
|type=&quot;{}&quot;}
9% = { 0.09_5 }

{
|type=&quot;{}&quot;}
9834% = { 98.34_5 }

{
|type=&quot;{}&quot;}
20% = { 0.2_5 }

{
|type=&quot;{}&quot;}
4% = { 0.04_5 }

{
|type=&quot;{}&quot;}
7.6% = { 0.076_5 }

{
|type=&quot;{}&quot;}
6% = { 0.06_5 }
&lt;/quiz&gt;
===== Decimal → Percentage =====
Now to convert a decimal into percentage we essentially do the complete opposite. We have &lt;math&gt;98.34&lt;/math&gt;. We need this to be a percentage (easier to read). Move the decimal point two places to the right. So, watch: &lt;math&gt;98.34&lt;/math&gt; → &lt;math&gt;983.4&lt;/math&gt; → &lt;math&gt;9834.&lt;/math&gt; --Now, we have &lt;math&gt;9834.&lt;/math&gt;, but the decimal point, since it's now a percentage, should not be there, but instead, a percentage should talk the decimal point's place. Now, we have our final result of &lt;math&gt;9834%&lt;/math&gt;.

==== Finding percent of a number ====
[[File:Universität Bonn.jpg|thumb|right|Would this be the fictional university these students were trying to get accepted to?]]
So, 95 students applied to a university (the fictional [[User:Atcovi/Mustafa Einhoonansebadoi University|Mustafa Einhoonansebadoi University]], for example), and only 20% of the students made it. 20%? What? With this in mind, we want to find &lt;math&gt;20%&lt;/math&gt; of &lt;math&gt;95&lt;/math&gt;. We take the percentage, &lt;math&gt;20%&lt;/math&gt;, and divide it by &lt;math&gt;100&lt;/math&gt;. So we get &lt;math&gt;20/100&lt;/math&gt; = &lt;math&gt;.2&lt;/math&gt;. Then, we multiply &lt;math&gt;.2&lt;/math&gt; by &lt;math&gt;95&lt;/math&gt;, in which we get &lt;math&gt;19&lt;/math&gt;. So &lt;math&gt;20%&lt;/math&gt; of &lt;math&gt;95&lt;/math&gt; is &lt;math&gt;19&lt;/math&gt;. Therefore, only 19 students out of 95 students made it into the fictional Mustafa Einhoonansebadoi University. 

{{subpage navbar}}
[[Category:Speak Math Now!]]</text>
      <sha1>fxyyo9wepgw08nuud6j6it29skyt4ru</sha1>
    </revision>
    <revision>
      <id>2807866</id>
      <parentid>2807865</parentid>
      <timestamp>2026-05-07T09:47:20Z</timestamp>
      <contributor>
        <username>~2026-27598-14</username>
        <id>3070872</id>
      </contributor>
      <origin>2807866</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="21636" sha1="ph7casbeu13m4grapbeeag5zv36ube9" xml:space="preserve">[[File:Quadratic formula.svg|thumb|right|An example of a Algebra formula (quadratic formula)]]
{{mathematics}}
{{secondary education}}
{{lesson}}
{{complete}}

'''Algebra''' (from the Arabic word &quot;al-jabr&quot; (الجبر), meaning &quot;reunion of broken parts&quot;) can feel like quite a complicated language of mathematics. However, as time goes on, completing Algebra will get easier and easier until it's a breeze. Completing Algebra takes true dedication with a worthwhile reward. This week, we will get into what Algebra is, and some warm ups (on arithmetic). Even though this may seem pointless, it is IMPORTANT that you review through these warm ups and get comfortable in solving them to lay a strong foundation for understanding larger topics later on.

Without further do, let's dig right into this!
==Algebra==
===What is Algebra?===
[[File:AlgebraJournalWork11-14-16.jpg|thumb|left|You might have to do this much work for a small answer!]]
In summary, Algebra solves for values that are not known yet. Such as □ - 4 = 6... instead of this empty box, we replace that with something known as a '''variable''', which is a letter that is used for something we do not know yet (of it's value). 

So, for the problem: □ - 4 = 6... we will have to do some math! We need to add positive 4, to the negative 4 on the left handside of the equation. When you add a number to another number (in which the solution becomes 0), you have cancelled out that number. But, you cannot just cancel out 4, and be done with it! You have to also add 4 to the number 6 on the righthand side of the equal sign, which brings out a golden rule in Algebra: '''To keep the balance, what we do to one side of the &quot;=&quot; we should also do to the other side!&quot;. ''

Now... once you have added 6 + 4 (which is 10), you should get □ = 10.

Great! You've done what's called &quot;solving for x&quot;, but in this case, it's an empty box... well guys, you're going to have to throw out that box, because it will be represented by a variable (a letter, from a - z). The letter that is commonly used for variables is x (and the reason for this  dates back to the origin of Algebra itself; Muhammad ibn Musa al-Khwarizmi, one of the main &quot;founders&quot; of Algebra you could say, used to call the unknown or &quot;box&quot; in our example, &quot;'''shay'''&quot;. &quot;'''Shay&quot;''' comes from the Arabic word '''شَيْء''', which essentially means &quot;thing&quot;. When Al-Khwarizmi's works were translated to Latin in medieval Spain, &quot;shay&quot; was translated as '''&quot;xay&quot;,''' since the letter x was pronounced as sh in Spain. Later on, this word &quot;'''xay&quot;''' got abbreviated to &quot;'''x&quot;''' to repesent the symbol of the unkown. Source/For more information, check out this [https://www.pbs.org/empires/islam/innoalgebra.html PBS] page). Brief history lesson aside, it's interesting to learn why &quot;x&quot; is the default norm when solving for the unkown in Algebra! '''Importantly, know that the unknown can take on any letter or variable name, not just x (i.e. could be a, b, c, y, even words if you wanted, etc.). Just remember what that variable is representing (the unknown value in this case).''' 

So instead of □ = 10, it's actually '''x = 10'''.

===== Sample problems of ''solving for x'' =====
&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{''x'' − 9 = 20
|type=&quot;{}&quot;}
''x''={ 29_3 }

{''x'' − 3 = 6
|type=&quot;{}&quot;}
''x''={ 9_3 }

{''x'' + 5 = 15
|type=&quot;{}&quot;}
''x''={ 10_3 }

{''x'' + 17 = 23
|type=&quot;{}&quot;}
''x''={ 6_3 }

{4''x'' = 12
|type=&quot;{}&quot;}
''x''={ 3_3 }

{''x''/2 = 0.5
|type=&quot;{}&quot;}
''x''={ 1_3 }

{''x''/50 = 2
|type=&quot;{}&quot;}
''x''={ 100_3 }

{''x''/9 = 5
|type=&quot;{}&quot;}
''x''={ 45_3 }
&lt;/quiz&gt;

Seems simple, huh? Well, it will get complicated, which is why it is important for you to do some review of your arithmetic! Let's dig into that...

==Arithmetic==
[[File:Multiply 4 bags 3 marbles.svg|thumb|right|4 x 3 = 12 (multiplication)]]
'''Arithmetic''' has to deal with elementary/basic levels of math, such as division, multiplication, subtraction, and addition. Basically, just working with numbers.

This SHOULD be a level familiar with you. If you are not familiar with arithmetic math/rules, then PLEASE review through Arithmetic, as you won't survive even the 1st step of Algebra. Trust me, the basics are THAT important.
===Fractions===
[[File:Cake quarters.svg|thumb|left|A Cake with fractions]]
'''Fractions''' (from Latin ''fractus'', &quot;broken&quot;) are parts of a whole. On the left side in the image of the cake, there is only &lt;math&gt;3/4&lt;/math&gt;'s of the cake showing, the other &lt;math&gt;1/4&lt;/math&gt; has been eaten/taken away.

The number, ''3'', in &lt;math&gt;3/4&lt;/math&gt;, is what is known as a '''numerator''' (Numerator: Number at the top, tells us of how much of the number is being talked about/being used). The number, ''4'', in &lt;math&gt;3/4&lt;/math&gt;, is what is known as a '''denominator''' (Denominator: Number showing the all time total).
;
;Simplest form/reduced form
A reduced form of a fraction is a fraction that cannot be divided by any number other than 1, and the denominator is greater than 1. So &lt;math&gt;2/4&lt;/math&gt; is NOT in simplest form, since we can divide 2 and 4, by 2... which results in the following number: &lt;math&gt;1/2&lt;/math&gt;. Though, not every fraction can be divided by 2, there are fractions, such as: &lt;math&gt;5/35&lt;/math&gt;, &lt;math&gt;7/21&lt;/math&gt;, and &lt;math&gt;30/5&lt;/math&gt;. The two first fractions are not divisible by 2, and &lt;math&gt;30/5&lt;/math&gt; can not be divided by 2 on both sides, but only on &lt;math&gt;30&lt;/math&gt;. It's important to simplify as if you were in a test, your teacher will mark your problems as incorrect if you didn't simplify your fractions. Keep in mind that simplifying a fraction into its simplest/reduced form doesn't change its value, both the original (unsimplified) fraction and its reduced form represent the same exact value/quantity. So, &lt;math&gt;\tfrac{2}{4}&lt;/math&gt; and &lt;math&gt;\tfrac{1}{2}&lt;/math&gt; represents the same quantity, a half! 

Here, we will present a few fractions for you to simplify.
====Sample problems for ''simplifying fractions'' (use ''/'' as the fraction line)====
&lt;quiz display=simple points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{6}{8}=&lt;/math&gt;{ 3/4_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{4}{60}=&lt;/math&gt;{ 1/15_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{30}{90}=&lt;/math&gt;{ 1/3_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{8}{18}=&lt;/math&gt;{ 4/9_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{9}{72}=&lt;/math&gt;{ 1/8_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{64}{46}=&lt;/math&gt;{ 32/23|1 9/23_7 }

{
|type=&quot;{}&quot;}
&lt;math&gt;\tfrac{206}{340}=&lt;/math&gt;{ 103/170_7 }
&lt;/quiz&gt;
===== Adding or Subtracting Fractions =====
[[File:Fractionsworkalgebra.PNG|thumb|right|What we just worked on, summarized]]
To simply add or subtract fractions, make sure the denominators of the fractions you are adding or subtracting are the same. If they are not, find the least common denominator (LCD). For example, if you want to add &lt;math&gt;\tfrac{4}{2}&lt;/math&gt; and &lt;math&gt;\tfrac{4}{6}&lt;/math&gt;, you first have to multiply the 2 in &lt;math&gt;\tfrac{4}{2}&lt;/math&gt; by '''3''', which equals '''6'''... BUT you cannot just multiply 2 only, you also have to multiply 4 by 3, since that's what you did to 2, the denominator. If you change the denominator, you have to change the numerator. ('''This step is crucial as it allows you to preserve the same value of the fraction''' but with just a different representation)

Alright, we got that out of the way, so once we have &lt;math&gt;\tfrac{12}{6}&lt;/math&gt; + &lt;math&gt;\tfrac{4}{6}&lt;/math&gt;, we can simply add. So &lt;math&gt;12&lt;/math&gt; + &lt;math&gt;4&lt;/math&gt; = &lt;math&gt;16&lt;/math&gt;, but don't add the denominators, they stay the same. So the answer is &lt;math&gt;\tfrac{16}{6}&lt;/math&gt;, and then we simplify down to &lt;math&gt;\tfrac{8}{3}&lt;/math&gt; dividing by 2 on both the numerator and denominator. 

But... did you notice something? &lt;math&gt;\tfrac{8}{3}&lt;/math&gt;? That doesn't seem right, does it? The denominator is smaller than the numerator. When you have a fraction like this, you have to convert it to a '''mixed fraction''' (skip to [[Speak_Math_Now!/Week_1:_Introduction_To_Algebra#Improper_Fraction_--.3E_Mixed_Fraction|section 2.1.1.4]]).

===== Multiplying Fractions =====
To multiply fractions, its easiest to first simplify your fraction to simplest terms. Once you have done that, you can simply multiply the numerators and the denominators. And obviously, simplify your final product, if you can. So, we have &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; and &lt;math&gt;\tfrac{2}{6}&lt;/math&gt;. You could multiply the numerators and denominators straight away and simplify at the end if you are comfortable, but to make it easier and clearer, we should simplify the fractions first. We simplify 6 and 8 by dividing both by 2, we also divide 2 and 6 by 2. So the fractions are now &lt;math&gt;\tfrac{3}{4}&lt;/math&gt; and &lt;math&gt;\tfrac{1}{3}&lt;/math&gt;. You simply multiply those two fractions by multiplying the numerator by the numerator, and doing the same for the denominators. After completing this process, you will get a solution (in fraction form). &lt;math&gt;\tfrac{3}{4}&lt;/math&gt; × &lt;math&gt;\tfrac{1}{3}&lt;/math&gt; &lt;math&gt;=&lt;/math&gt; &lt;math&gt;\tfrac{3}{12}&lt;/math&gt;. 

&lt;math&gt;\tfrac{3}{12}&lt;/math&gt; is not going to be our final product, though, since we can simplify the fraction by dividing the fraction by 3, which results in &lt;math&gt;\tfrac{1}{4}&lt;/math&gt;.

===== Dividing Fractions =====
There is an interesting twist when it comes to dividing fractions. You have to turn the fraction you want to divide by (second fraction) upside-down, also known as &quot;Keep, Change, Flip&quot; where you keep the first fraction the same, change the operation to multiplication, and replace the second fractions numerator with the denominator and the denominator with the numerator. Not only that, you have to turn the division symbol (÷) into a multiplication symbol (× or •). After that, you use your skills you learned in multiplying a fraction, and you multiply both of the fractions. Simplify if you need to.

So, &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; ÷ &lt;math&gt;\tfrac{7}{12}&lt;/math&gt;. Change the division symbol to a multiplication symbol, and turn the fraction you want to divide by upside-down (the upside-down fraction is known as a '''reciprocal'''). So &lt;math&gt;\tfrac{6}{8}&lt;/math&gt; × (or •) &lt;math&gt;\tfrac{12}{7}&lt;/math&gt;. Multiply the numerators and denominators. The answer is &lt;math&gt;\tfrac{72}{56}&lt;/math&gt;, simplified down to &lt;math&gt;\tfrac{9}{7}&lt;/math&gt;.

===== Improper Fraction --&gt; Mixed Fraction =====
Divide the numerator by the denominator. The '''quotient''' (result of the division taking place/number above the division line) will be the whole number of the mixed fraction, while the numerator will be the remainder. The denominator remains unchanged, so don't change the denominator at all!

{{notice|If you would like to take the quiz on Fractions, please go to '''[[Speak Math Now!/Week 1: Introduction To Algebra/Fractions Quiz]]'''}}

See also: https://www.tes.com/lessons/bJieZ4sFPJbSTw/fractions-4-mixed-numbers-and-improper-fractions

===Decimals===

Ever wondered how to write 8&lt;math&gt;\tfrac{47}{100}&lt;/math&gt; as a decimal? Well, you've got the answer: 8.47! How did we get that answer? Let's look at a few more and maybe you'll see the pattern:
# 6&lt;math&gt;\tfrac{98}{100}&lt;/math&gt; = 6.98
# 2&lt;math&gt;\tfrac{56}{100}&lt;/math&gt; = 2.56
# 9&lt;math&gt;\tfrac{27}{100}&lt;/math&gt; = 9.27
# 5&lt;math&gt;\tfrac{83}{100}&lt;/math&gt; = 5.83
You see? We simply put the mixed number in front of the dot, and with the numerator, we slap that behind the dot! Throw out the 100, it's not important when building your decimal. 

Decimals are all about place value, the value of a number in a specific place in a number. So, when we have &lt;math&gt;6.72&lt;/math&gt;, the &lt;math&gt;6&lt;/math&gt; is in the Ones place. Now, let's throw &lt;math&gt;9&lt;/math&gt; in the tens place, which is 10 times bigger than the Ones place: &lt;math&gt;96.72&lt;/math&gt;. But... that's doesn't seem enough, does it? Let's throw in a &lt;math&gt;6, 2, 8&lt;/math&gt; and a &lt;math&gt;3&lt;/math&gt; in there! And now, we have: &lt;math&gt;628,396.72&lt;/math&gt;.

Woah! That's a pretty big number, but we can easily break this number down to it's place value. Let's do it!

So, our number, &lt;math&gt;628,396.72&lt;/math&gt;, is the number we need to break down. Let's start from the decimal point, and move left:
* The number &lt;math&gt;6&lt;/math&gt; is in the Ones place. '''x10'''
* The number &lt;math&gt;9&lt;/math&gt; is in the Tens place. '''x10'''
* The number &lt;math&gt;3&lt;/math&gt; is in the Hundreds place. '''x10'''
* The number &lt;math&gt;8&lt;/math&gt; is in the Thousands place. '''x10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Ten thousands place. '''x10'''
* The number &lt;math&gt;6&lt;/math&gt; is in the Hundred Thousands place.

Now we have broken up the numbers left of the decimal--What about the numbers on the ''right''? Let's throw in a &lt;math&gt;5, 2, 4&lt;/math&gt; and a &lt;math&gt;7&lt;/math&gt;. Now, we have &lt;math&gt;628,396.725,247&lt;/math&gt;. Let's break this number up like we did above.

So, our number, &lt;math&gt;628,396.725,247&lt;/math&gt;, is the number we need to break down. This time, we need to start on the decimal point, and move ''right'':
* The number &lt;math&gt;7&lt;/math&gt; is in the Tenths place. '''x-10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Hundredths place. '''x-10'''
* The number &lt;math&gt;5&lt;/math&gt; is in the Thousandths place. '''x-10'''
* The number &lt;math&gt;2&lt;/math&gt; is in the Ten Thousandths place. '''x-10'''
* The number &lt;math&gt;4&lt;/math&gt; is in the Hundred Thousandths place. '''x-10'''
* The number &lt;math&gt;7&lt;/math&gt; is in the Millionths place. 

We have just now gone over the importance of Place Value in the Decimal World. Now, we will go into how to work with decimals, in the Decimal World!

See also: http://www.shmoop.com/fractions-decimals/place-value-naming-decimals.html

==== Adding/Subtracting Decimals ====
To add decimals, in addition column-style, put the decimals in its place with the decimals lined up. Then simply add on. So, for &lt;math&gt;1.5&lt;/math&gt; + &lt;math&gt;2.5&lt;/math&gt; we'd line up the decimal points. But, if we had a problem like &lt;math&gt;1.15&lt;/math&gt; + &lt;math&gt;2.0&lt;/math&gt;, we'd add a &lt;math&gt;0&lt;/math&gt; after the &lt;math&gt;0&lt;/math&gt; that is behind the decimal. Adding a zero to a place in a decimal means &quot;no value&quot;. So &lt;math&gt;10&lt;/math&gt; basically means no ones, and &lt;math&gt;100&lt;/math&gt;, means no ones or hundreds. Same things goes for subtracting as well folks.
=====Sample problems for ''adding/subtracting decimals''=====
&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
6.8 - 2.5 = { 4.3_6 }

{
|type=&quot;{}&quot;}
3.4 + 5.6 = { 9_6 }

{
|type=&quot;{}&quot;}
9 + 4.50 = { 13.5_6 }

{
|type=&quot;{}&quot;}
41.89 + 25.00 = { 66.89_6 }

{
|type=&quot;{}&quot;}
9.01 + 3.089 = { 12.099_6 }

{
|type=&quot;{}&quot;}
10.90 + 11.1 = { 22_6 }

{
|type=&quot;{}&quot;}
9.5 + 3.44 = { 12.94_6 }

{
|type=&quot;{}&quot; coef=&quot;2.5&quot;}
9.00 x 2.00 = { 18_6 } &lt;big&gt;(BONUS!)&lt;/big&gt; 
&lt;/quiz&gt;

==== Multiplying Decimals ====
[[File:9.82x5.73 multiplication image.svg|thumb|A visual representation of the multiplication example]]
Multiplying decimals isn't as hard as it really seems to be. So, we have &lt;math&gt;9.83&lt;/math&gt; × &lt;math&gt;5.73&lt;/math&gt;. For most people, column multiplication is a lot easier than side-by-side multiplication. That being mentioned, let us column these numbers:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;× &lt;math&gt;5.73&lt;/math&gt;
-------

Now that we have our problem, we should simply ignore the decimal points and just multiply as usual, so you should get this answer once you are done with that (remember to add a zero (and grow with zeros in each line) to each and every line of addition):

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;        × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt; &lt;br&gt;  &lt;math&gt;+&lt;/math&gt;   &lt;math&gt;68810&lt;/math&gt; &lt;br&gt;  &lt;math&gt;+&lt;/math&gt; &lt;math&gt;491500
&lt;/math&gt;
-------

With the simple usage of addition, we should get:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;        × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt;&lt;br&gt;   &lt;math&gt;+&lt;/math&gt;  &lt;math&gt;68810&lt;/math&gt;&lt;br&gt;  &lt;math&gt;+
&lt;/math&gt; &lt;math&gt;491500&lt;/math&gt;
-------
&lt;math&gt;563259&lt;/math&gt;

Now, we need to bring back our handy dandy decimal point, but where? In &lt;math&gt;9.83&lt;/math&gt; and &lt;math&gt;5.73&lt;/math&gt;, there are FOUR numbers in these 2 numbers overall that are behind the decimal point (in each number, there are two numbers behind the decimal points). So, we have &lt;math&gt;9.83&lt;/math&gt; and &lt;math&gt;5.73&lt;/math&gt;. Now, that totals up to four numbers overall behind the decimal point. So in &lt;math&gt;563259&lt;/math&gt;, we need to move the decimal point four times (beginning from the right). So watch as follows:

&lt;math&gt;563259.&lt;/math&gt;&lt;br&gt;
&lt;math&gt;56325.9&lt;/math&gt;&lt;br&gt;
&lt;math&gt;5632.59&lt;/math&gt;&lt;br&gt;
&lt;math&gt;563.259&lt;/math&gt;&lt;br&gt;
&lt;math&gt;56.3259&lt;/math&gt;

That simple. Now, review your work, your whole work should look like this:

&lt;math&gt;9.83&lt;/math&gt;&lt;br&gt;       × &lt;math&gt;5.73&lt;/math&gt;
-------
&lt;math&gt;2949&lt;/math&gt;&lt;br&gt;    &lt;math&gt;+&lt;/math&gt;&lt;math&gt;68810&lt;/math&gt;&lt;br&gt;  &lt;math&gt;+&lt;/math&gt;&lt;math&gt;491500&lt;/math&gt;
-------
&lt;math&gt;56.3259&lt;/math&gt;

==== Dividing Decimals ====
;Dividing a decimal by a whole number
If you want to divide a decimal by a whole number, you should divide the 2 numbers, omitting the decimal point. After you are done dividing, add the decimal point to the '''quotient''' (final product/answer at the top of the long division symbol). The decimal should be right above the decimal point in the '''dividend''' (number in the box/number that is being divided). It's quite easy and simple, as long as you know how to do long division and if you are still familiar with long division. 

Hey, this seems ''too'' easy--Let's figure out how to divide a decimal by a decimal!
;Dividing a decimal by a decimal
The trick to dividing a decimal by a decimal is to shift the decimal point as many times as it gets to a whole number, so follow along: &lt;math&gt;69.45&lt;/math&gt; ÷ &lt;math&gt;5.78&lt;/math&gt;. Now, we simply move the decimal point as many times as we need to make the number we are going to use to divide 69.45 a whole number, so watch as followed:&lt;br&gt;
&lt;math&gt;69.45&lt;/math&gt; ÷ &lt;math&gt;5.78&lt;/math&gt; →&lt;br&gt;
&lt;math&gt;694.5&lt;/math&gt; ÷ &lt;math&gt;57.8&lt;/math&gt; →&lt;br&gt;
&lt;math&gt;6945&lt;/math&gt;. ÷ &lt;math&gt;578&lt;/math&gt;.


Now that we have finally got our dividend a whole number (and now our first number that we are going to divide), we can go ahead and divide normally (using long division). In the end, &lt;math&gt;69.45&lt;/math&gt; divided by &lt;math&gt;5.78&lt;/math&gt; should get you &lt;math&gt;12.0155709&lt;/math&gt;!

A pretty simple one we could go is &lt;math&gt;6.4&lt;/math&gt; ÷ &lt;math&gt;0.4&lt;/math&gt;, here, we simply move our dots like so:&lt;br&gt;
&lt;math&gt;6.4&lt;/math&gt; ÷ &lt;math&gt;0.4&lt;/math&gt;&lt;br&gt;
&lt;math&gt;64&lt;/math&gt; ÷ &lt;math&gt;04.&lt;/math&gt;&lt;br&gt;
&lt;math&gt;64&lt;/math&gt; ÷ &lt;math&gt;4&lt;/math&gt;&lt;br&gt;
Then, we can simply divide, heck... we don't even need to do long division! The answer should pop in your head, which is &lt;math&gt;16&lt;/math&gt;.

{{notice|If you would like to take the quiz on Decimals, please go to '''[[Speak Math Now!/Week 1: Introduction To Algebra/Decimals Quiz]]'''}}

===Percentages===
A good definition of &quot;percent&quot; is a fraction in which the denominator is the number &lt;math&gt;100&lt;/math&gt;. For example, the numbers &lt;math&gt;59%&lt;/math&gt;, &lt;math&gt;63%&lt;/math&gt;, &lt;math&gt;91%&lt;/math&gt;, and &lt;math&gt;85%&lt;/math&gt;, are the same as just saying &lt;math&gt;\tfrac{59}{100}&lt;/math&gt;, &lt;math&gt;\tfrac{63}{100}&lt;/math&gt;, &lt;math&gt;\tfrac{91}{100}&lt;/math&gt;, and &lt;math&gt;\tfrac{85}{100}&lt;/math&gt;. You could also say 59 out of 100 parts, 63 out of 100 parts, 91 out of 100 parts, and 85 out of 100 parts.
====Converting Percentages====
Now that we got the basis of percentages and how they operate, we should look into changing percentages.
===== Percentage → Decimal =====
Let's look in turning a percentage into a decimal point first. It's very simple. Let's say you have &lt;math&gt;\tfrac{9}{100}&lt;/math&gt;, which, in percentage form, is &lt;math&gt;9%&lt;/math&gt;. So, we have 9%. Now, we want to change it to a decimal (I don't know, think of a reason). We simply convert the percentage symbol into a decimal point, so like this: &lt;math&gt;9.&lt;/math&gt;. Now, we have &lt;math&gt;9.&lt;/math&gt;, so then we move the decimal number two places to the left, like so: &lt;math&gt;9.&lt;/math&gt; → &lt;math&gt;.9&lt;/math&gt; → &lt;math&gt;.09&lt;/math&gt;. So now, we have &lt;math&gt;0.09&lt;/math&gt;. We added the 2 zeros in because there is no value in the tenths place, and because &lt;math&gt;.09&lt;/math&gt; does not look quite right. Looks a bit off.
===== Samples problems for ''converting percentages to decimals'' =====
&lt;quiz display=&quot;simple&quot; points=&quot;1/1&quot;&gt;
{
|type=&quot;{}&quot;}
59% = { 0.59_5 }

{
|type=&quot;{}&quot;}
63% = { 0.63_5 }

{
|type=&quot;{}&quot;}
91% = { 0.91_5 }

{
|type=&quot;{}&quot;}
85% = { 0.85_5 }

{
|type=&quot;{}&quot;}
9% = { 0.09_5 }

{
|type=&quot;{}&quot;}
9834% = { 98.34_5 }

{
|type=&quot;{}&quot;}
20% = { 0.2_5 }

{
|type=&quot;{}&quot;}
4% = { 0.04_5 }

{
|type=&quot;{}&quot;}
7.6% = { 0.076_5 }

{
|type=&quot;{}&quot;}
6% = { 0.06_5 }
&lt;/quiz&gt;
===== Decimal → Percentage =====
Now to convert a decimal into percentage we essentially do the complete opposite. We have &lt;math&gt;98.34&lt;/math&gt;. We need this to be a percentage (easier to read). Move the decimal point two places to the right. So, watch: &lt;math&gt;98.34&lt;/math&gt; → &lt;math&gt;983.4&lt;/math&gt; → &lt;math&gt;9834.&lt;/math&gt; --Now, we have &lt;math&gt;9834.&lt;/math&gt;, but the decimal point, since it's now a percentage, should not be there, but instead, a percentage should talk the decimal point's place. Now, we have our final result of &lt;math&gt;9834%&lt;/math&gt;.

==== Finding percent of a number ====
[[File:Universität Bonn.jpg|thumb|right|Would this be the fictional university these students were trying to get accepted to?]]
So, 95 students applied to a university (the fictional [[User:Atcovi/Mustafa Einhoonansebadoi University|Mustafa Einhoonansebadoi University]], for example), and only 20% of the students made it. 20%? What? With this in mind, we want to find &lt;math&gt;20%&lt;/math&gt; of &lt;math&gt;95&lt;/math&gt;. We take the percentage, &lt;math&gt;20%&lt;/math&gt;, and divide it by &lt;math&gt;100&lt;/math&gt;. So we get &lt;math&gt;20/100&lt;/math&gt; = &lt;math&gt;.2&lt;/math&gt;. Then, we multiply &lt;math&gt;.2&lt;/math&gt; by &lt;math&gt;95&lt;/math&gt;, in which we get &lt;math&gt;19&lt;/math&gt;. So &lt;math&gt;20%&lt;/math&gt; of &lt;math&gt;95&lt;/math&gt; is &lt;math&gt;19&lt;/math&gt;. Therefore, only 19 students out of 95 students made it into the fictional Mustafa Einhoonansebadoi University. 

{{subpage navbar}}
[[Category:Speak Math Now!]]</text>
      <sha1>ph7casbeu13m4grapbeeag5zv36ube9</sha1>
    </revision>
  </page>
  <page>
    <title>Social Victorians/People/Keppel</title>
    <ns>0</ns>
    <id>263976</id>
    <revision>
      <id>2807818</id>
      <parentid>2807555</parentid>
      <timestamp>2026-05-06T15:10:21Z</timestamp>
      <contributor>
        <username>Scogdill</username>
        <id>1331941</id>
      </contributor>
      <origin>2807818</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="30260" sha1="am94nnp6miruwwvlnpyhesfr2uya3ak" xml:space="preserve">== Also Known As ==
Freddie Keppel
*Family name: his, Keppel; hers, Edmunstone
*Freddie (Alice Frederica) Edmonstone Keppel
*Alice Keppel: [[viaf:62357923/|VIAF: 62357923]]
The Honourable George Keppel

Sir Derek Keppel

== Overview ==
[[File:Alice Keppel.jpg|thumb|alt=Painted portrait of the upper third of a woman wearing a white formal dress and large jewels|Alice Keppel, 1890s]]

Alice Keppel is shown (right) at about 30 years old. She died (at nearly 80) of cirrhosis of the liver.&lt;ref name=&quot;:0&quot; /&gt; George Keppel was the 3rd son of the Earl of Albemarle. Alice and George Keppel are the great grandparents of Queen Camilla, who was less than a year old when they died. Alice Keppel was one of two last mistresses of [[Social Victorians/People/Albert Edward, Prince of Wales|Albert Edward, Prince of Wales]], for many an improvement over [[Social Victorians/People/Warwick|Daisy, Countess of Warwick]], who was not discreet and who had enemies. Agnes Keyser seems to have begun a relationship with the Prince of Wales about the same time Alice Keppel did — around 1898. (Keyser was not a socialite, and Lamont-Brown says she mothered him in a way he had craved his entire life.&lt;ref name=&quot;:6&quot;&gt;Lamont-Brown, Raymond. ''Alice Keppel and Agnes Keyser: Edward VII's Last Loves''. History Press, 2011. [Preview on Google Books: [https://books.google.com/books?id=8LQTDQAAQBAJ&amp;source=gbs_navlinks_s https://books.google.com/books?id=8LQTDQAAQBAJ].&lt;/ref&gt;)

Famous for her tact, discretion and social skills, Alice Keppel and [[Social Victorians/People/de Soveral|Luís de Soveral]] were friends, and [[Social Victorians/People/Alexandra, Princess of Wales|Alex, Princess of Wales]], tolerated her and allowed her to visit King Edward VII on his deathbed, the only mistress granted this access.

In his account of the &quot;Social and Diplomatic Life&quot; of Edward VII, Gordon Brook-Shepherd describes ways George and Alice Keppel's lives changed after the Prince of Wales's accession to the throne:&lt;blockquote&gt;For Mrs Keppel, in particular, the accession brought greater problems as well as greater privileges. It was one thing to have an Heir-Apparent for a lover but something quite different when that lover became the ruler of the British Empire. To begin with, even to move in the appropriate style at the King's side cost a great deal more money, / and money was something that neither Edward VII nor the Hon. George Keppel had to spare. Indeed, in an attempt to put extra cash into the Keppel family coffers after Alice had become the mistress of a king, her husband was obliged to go &quot;into trade.&quot; Sir Thomas Lipton, the grocer millionaire and yachting friend of King Edward's, found a job for him in his &quot;Buyers' Association&quot; at No. 70–74 Wigmore Street. This, to judge from the firm's stationery on which George Keppel once wrote a business letter to [[Social Victorians/People/de Soveral|Soveral]], sold everything direct to the customer, from groceries, bedding and tobacco, to cartridges and coal. It also advertised &quot;Motor Cars Bought, Sold or Exchanged,&quot; and it was about this that Keppel, prompted by his Alice, wrote to the Portuguese Minister:

:&quot;Dear Soveral, My wife tells me you contemplate buying a small motor car for use in London. May we offer our services in the matter ...? [sic]

For an earl's son to be a salesman in Edwardian England was bad enough. For the salesman to be the husband of the King's official mistress was an added humiliation. Though George Keppel seems to have taken the whole situation philosophically, there were many in society who condemned him for being so much the ''mari complaisant''. As one distinguished survivor from that Edwardian age, who shall be anonymous, commented: &quot;Had Keppel been put up for membership at some London clubs, the black balls would have come rolling out like caviare.&quot;&lt;ref&gt;Brook-Shepherd, Gordon. ''Uncle of Europe: The Social and Diplomatic Life of Edward VII''. London: Collins, 1975. Internet Archive: [https://archive.org/details/uncleofeurope0000unse/page/62/mode/2up?q=soveral https://archive.org/details/uncleofeurope0000unse/].&lt;/ref&gt;{{rp|138–139}}&lt;/blockquote&gt;While her reputation was one of discretion, she was widely known to be the King's mistress and &quot;when it came to visits to the homes of the great and good with the king,&quot;&lt;blockquote&gt;she was always to be found standing or sitting near her lover in the official photographs for the picture papers of the day. And her &quot;pushy presence&quot;, some were to say, was such a constant irritation to Queen Alexandra that it drove her to eschew her husband's company.&lt;ref name=&quot;:6&quot; /&gt;&lt;/blockquote&gt;According to Lamont-Brown,&lt;blockquote&gt;Reginald Baliol Brett, 2nd Viscount Esher (1852–1930), royal archivist and intimate of every prime minister from Rosebery to Baldwin, ... believed that she deliberately lied in Society about certain royal happenings to enhance her own reputation.&lt;ref name=&quot;:6&quot; /&gt;&lt;/blockquote&gt;On 10 March 1932, Thursday, Virginia Woolf wrote in her diary about a lunch with Alice Keppel:&lt;blockquote&gt;I had lunched with Raymond [Mortimer, critic] to meet Mrs Keppel; a swarthy thick set raddled direct — &quot;My dear&quot;, she calls one — old grasper: whose fists had been in the moneybags these 50 years: And she has a flat in the Ritz; old furniture; &amp;c. I like her on the surface of the old courtezan: who has lost all bloom; &amp; acquired a kind of cordiality, humour, directness instead. No sensibilities as far as I could see; no snobberies, immense superficial knowledge, &amp; going to Berlin to hear Hitler speak. Shabby under dress: magnificent furs, great pearls: a Rolls Royce waiting. [sic editorial interpolation]&lt;ref name=&quot;:6&quot; /&gt;&lt;/blockquote&gt;

== Acquaintances, Friends and Enemies ==

=== Freddie Keppel's Friends ===

* [[Social Victorians/People/de Soveral|Luís de Soveral]]
* [[Social Victorians/People/Sarah Spencer-Churchill Wilson|Lady Sarah (Spencer-Churchill) Wilson]]&lt;ref name=&quot;:0&quot; /&gt;

=== Freddie Keppel's Sexual Partners Outside Her Marriage ===
* Ernest Beckett, 2nd [[Social Victorians/People/Grimthorpe|Baron Grimthorpe]]
* Humphrey Sturt, 2nd [[Social Victorians/People/Alington|Baron Alington]]
* [[Social Victorians/People/Albert Edward, Prince of Wales | Albert Edward, Prince of Wales]]

== Organizations ==

* The inner circle of the [[Social Victorians/People/Albert Edward, Prince of Wales|Prince of Wales and King Edward VII]] of England after his accession
** [[Social Victorians/People/Alexandra, Princess of Wales|Alexandra, Princess of Wales and Queen Alexandra]]
** [[Social Victorians/People/de Soveral|Luìs, Marquis de Soveral]]
** Alice Keppel
*&quot;Committee of Seven&quot;
*#Alice Keppel (&quot;gradually became chief of the 'Committee of Seven'&quot;&lt;ref name=&quot;:6&quot; /&gt;)
*#[[Social Victorians/People/de Soveral|Luìs, Marquis de Soveral]]
*#Sir [[Social Victorians/People/Cassel|Ernest Cassel]]
*#Sir William Esher&lt;ref name=&quot;:6&quot; /&gt;
*#Admiral Lord Fisher
*#Lord Hardinge
*#Sir Francis Knollys

== Timeline ==
'''1891 June 1''', Alice [[Social Victorians/People/Edmonstone|Edmonstone]] and George Keppel married.&lt;ref&gt;&quot;Alice Frederica Edmonstone.&quot; {{Cite web|url=http://www.thepeerage.com/p1723.htm#i17228|title=Person Page|website=www.thepeerage.com|access-date=2020-10-05}}&lt;/ref&gt;

'''1893 July 13, Thursday''', George and Alice Keppel attended the [[Social Victorians/Timeline/1893#The Countess of Listowel's Garden Party|Countess of Listowel's Garden Party]] [[Social Victorians/Timeline/1893#The Countess of Listowel's Garden Party|at her residence, Kingston House, Princes-gate]].

'''1893 December 12, Tuesday''', Alice and George Keppel took part in [[Social Victorians/Timeline/1893#12 December 1893, Tuesday|tableaux vivants at the Newland Bazaar in Hull]]. They were in [[Social Victorians/People/Arthur Stanley Wilson|Mrs. Arthur Wilson]]'s party.

'''1894 July 19, Thursday''', the Hon. George and Alice Keppel were guests at the ball following a [[Social Victorians/Timeline/1894#19 July 1894, Thursday|dinner hosted by the Duke and Duchess of Devonshire for the Prince and Princess of Wales]] and their family.

'''1895 February 1, Friday''', the Hon. George Keppel and Alice Keppel attended the [[Social Victorians/1895 Bal Poudre Warwick Castle|bal poudré at Warwick castle]].

'''1896 April 27, Monday''', Alice Keppel gave a box to [[Social Victorians/Timeline/1896#27 April 1896, Monday|Lady Angela St Clair Erskine and James Stewart Forbes for their wedding]].

'''1897 June 28, Monday''', the Hon. George and Alice Keppel attended [[Social Victorians/Diamond Jubilee Garden Party|the Queen's Garden Party at Buckingham Palace]], the last official event of the Diamond Jubilee. They were two of perhaps 5,000 or 6,000 people present.

'''1897 July 2''', Alice and George Keppel attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House. Freddie Keppel's brother and sister-in-law, [[Social Victorians/People/Edmonstone|Sir Archibald and Lady Edmonstone]], also attended.

'''1897 July 6, Tuesday''', Alice and George Keppel attended a [[Social Victorians/Timeline/1897#6 July 1897, Tuesday|garden party at Devonshire House]]. No one from Victoria's or the Prince of Wales's family was present, although a number of dignitaries from around the empire were.

'''1897 July 31, Saturday''', Alice and George Keppel may have attended the [[Social Victorians/Timeline/1897#31 July 1897, Saturday|wedding of Mabel Caroline Wombwell and Henry R. Hohler]] and the reception afterwards, although their names are not listed. The ''Morning Post'' does list a gift from the Keppels' — &quot;white enamel and turquoise sleeve links.&quot;&lt;ref&gt;&quot;Marriage of Mr. H. R. Hohler and Miss Wombwell.&quot; ''Morning Post'' 2 August 1897, Monday: 6 [of 8], Col. 3a–c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970802/067/0006 (accessed June 2019).&lt;/ref&gt; The [[Social Victorians/People/Albert Edward, Prince of Wales|Prince]] and [[Social Victorians/People/Alexandra, Princess of Wales|Princess of Wales]] did not attend or send a gift.

'''1897 November 20, Saturday or so''', [[Social Victorians/Timeline/1897#20 November 1897, Saturday|house parties for Derby horseraces run at Epsom Downs]]. The ''Derby Mercury'' cites the ''Daily Mail'': unlike 10 years ago, &quot;now all the smartest people go, and it is one of the most important meetings, rivalling Doncaster in popularity.&quot; The Keppels were guests at the Miller Mundy house party at Shipley Hall: &quot;At Shipley are Sir Charles and Lady Hartopp, Mr. and Mrs. George Keppel, Mrs. de Winton, Lord Athlumney, and Mr. Sturt, among others.&quot;&lt;ref&gt;&quot;Hints for Ladies. Fashion at Derby Races.&quot; ''Derby Mercury'' 24 November 1897, Wednesday: 6 [of 8], Col. 5a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000052/18971124/050/0006.&lt;/ref&gt;

'''1898''', Alice Keppel became the mistress of [[Social Victorians/People/Albert Edward, Prince of Wales |Albert Edward, Prince of Wales]]. The article on her in ''Wikipedia''&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|date=2020-06-11|title=Alice Keppel|url=https://en.wikipedia.org/w/index.php?title=Alice_Keppel&amp;oldid=962041251|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Alice_Keppel.&lt;/ref&gt; says they met in 1898, but they both were present at the Duchess of Devonshire's ball in 1897, so she likely was presented if not introduced to him; on the other hand, more than 700 people were there.

'''1899 January 25, Wednesday''', the Hon. George Keppel, at least, and perhaps Freddie, was present at the [[Social Victorians/Timeline/1899#25 January 1899, Wednesday|Holderness Hunt Ball]].

'''1899 July 5, Wednesday''', George and Alice Keppel attended a [[Social Victorians/Timeline/1899#Dinner and Dance at Devonshire House|dance at Devonshire House hosted by the Duke and Duchess of Devonshire]].

'''1899 July 6, Thursday''', George and Alice Keppel attended the [[Social Victorians/Timeline/1899#Joan Wilson and Guy Fairfax's Wedding|wedding of Joan Wilson and Guy Fairfax in St. Mark's, near Grosvenor Square]]. Their daughter Violet was train-bearer.

'''1899 September 4, Monday,''' the Hon. George and Mrs. George Keppel were at [[Social Victorians/People/Holden#1899 September 4, Monday|the house party of Mr. E. W. Meckett, M.P., at Kirkstall Grange]] that week, held as part of the Doncaster races.

'''1900 July 27, Friday''', Alice Keppel (at least) was present at a [[Social Victorians/People/Arthur Stanley Wilson#1900 July 27, Friday|dinner party for Albert Edward, Princes of Wales hosted by the Arthur Wilsons]]. George Keppel's name is not listed.

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==
Freddie Keppel — the Hon. Mrs. George Keppel — attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] dressed as Madame de Polignac. The Hon. George Keppel attended, dressed as King Solomon. Neither appears to have been in the first supper seating, suggesting that her prominence was yet to come. Also, they did not use any of the most notable costumiers (like [[Social Victorians/People/Dressmakers and Costumiers#Mr. Alias|Mr. Alias]]) or couturiers (like [[Social Victorians/People/Dressmakers and Costumiers#Mrs. Mason|Mrs. Mason]]) who sometimes provided the press with information about or perhaps viewings of the costumes. Further, their portraits do not appear in the commemorative album. The newspapers commented on the Keppels' costumes, but no portrait of either for this ball survives.[[File:Duchess of Polignac by E.Vigee-Lebrun (1787, Atheneum).jpg|thumb|alt=Painting of the upper half of a woman wearing a white dress with a black lacy shawl and a large straw hat over hair that is loose and lightly powdered|''Duchess of Polignac'' by Vigee-Le Brun, 1787]]
=== The Hon. Mrs. George Keppel ===
The portrait of Gabrielle, Duchess of Polignac (right), painted by Élisabeth Louise Vigée Le Brun in 1787, shows the duchess at about 30 years old, about what Alice Keppel was at the time of the ball. Vigée Le Brun painted several portraits of Gabrielle around this time, and in them all, Gabrielle appears informally dressed, without jewelry, so the portraits seem more intimate than official. Gabrielle's dress in this portrait is not the original for Keppel's dress — its style is transitional between the stiff fabrics and panniers of the formal court (and the past) and the post-revolutionary empire waists, columnar shape and lightweight fabrics. [[Social Victorians/People/Working in Publishing#Journalists|Ardern Holt]]'s description of Keppel's dress for ''The Queen'' (below) bears no relation to what we can see in Le Brun's portrait of Gabrielle, Duchess of Polignac.

Unlike this more informal presentation of Madame de Polignac, Freddie Keppel's costume appears to have been elaborate and pre-Revolutionary.

==== Newspaper Reports of Her Costume ====
Alice Keppel walked in the [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#Louis XV and XVI Period|Louis XV procession]] led by [[Social Victorians/People/Warwick|Daisy, Countess of Warwick]], who was dressed as Marie Antoinette.
*She was &quot;very beautiful,&quot; and, &quot;as Madame de Polignac, wore a lovely dress of silver cloth embroidered in silver and pink gems and garlands of small roses. The bunched out over dress was of pink and silver brocade lined with apple-green satin.&quot;&lt;ref&gt;&quot;Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses.&quot; London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and http://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.&lt;/ref&gt;{{rp|p. 6, Col. 1a}}
*She was dressed as Madame de Polignac in the quadrille of the Louis XV and XVI Period.&lt;ref name=&quot;:1&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;
*&quot;Mrs. George Keppel, as Madame de Polignac, wore a dress of silver cloth embroidered in silver and pink gems and garlands of small roses. The bunched-out over-dress was of pink and silver brocade lined with apple-green satin.&quot;&lt;ref name=&quot;:3&quot; /&gt;{{rp|p. 3, Col. 3c}}
*&quot;Among these [in the Countess of Warwick's Marie Antoinette quadrille] the Honourable Mrs. George Keppell looked very beautiful.... Mrs. George Keppell, as Madame de Polignac, wore a lovely dress of silver cloth embroidered in silver, and pink gems and garlands of small roses. The bunched-out over-dress was of pink and silver brocade lined with apple green satin.&quot;&lt;ref&gt;&quot;The Duchess of Devonshire's Fancy Dress Ball. Special Telegram.&quot; ''Belfast News-Letter'' Saturday 03 July 1897: 5 [of 8], Col. 9 [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0000038/18970703/015/0005.&lt;/ref&gt;{{rp|p. 5, Col. 9c}}
*Ardern Holt's writing for ''The Queen'' is typically more trustworthy on fashion and garment construction than other, more traditional news outlets: &quot;''Madame de Polignac'' appeared at the Devonshire House ball in the Louis XV. quadrille, represented by the Hon. Mrs George Keppel, in an old dress of the period, a rose and silver brocade handed down from that century. The [[Social Victorians/Terminology#Hoops|pouf paniers]] [sic] were lined with pale green soft satin, and faced back with wide bands of silver embroidery. The hooped petticoat was of cloth of silver worked in tinsel threads of all shades, forming a design of roses in true lovers' knots extending from the waist to the hem; at the foot it was garlanded with pink pompon [sic] roses, interlaced through the stripes of embroidery, and below this was a quaint pleating of silver lace. The low bodice was finished off with a transparent lace collar sewn with silver. The hair was powdered and dressed very high, with soft curls falling on the neck, surmounted by a pink and green ostrich feather and a small garland of roses. She wore long lace mittens.&quot;&lt;ref&gt;Holt, Ardern. “Fancy Dress.” ''The Queen'' 31 July 1897, Saturday: 43 [of 84], Col. 3c [of 3]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18970731/280/0043.&lt;/ref&gt;

==== Commentary on the Costume ====
This costume has striking similarities to a costume Freddie Keppel wore to the Countess of Warwick's 1895 ''bal poudré'', suggesting that perhaps the 1895 dress was very successful or that the Keppels were trying not to spend extravagantly.

* The gown is pink in both descriptions, and made of antique fabric.
* Both costumes were decorated with roses, embroidered as well as garlanded. In 1895, both the gown and the petticoat were embroidered with roses, the petticoat with garlands of roses. The 1897 petticoat has roses on the petticoat in a striped embroidered (perhaps a woven fabric instead?) pattern as well as roses made of pink pompoms.
* The petticoats were very different. The 1897 petticoat was &quot;cloth of silver&quot; and the 1895 petticoat was of &quot;dull creamy-tinted satin.&quot;
* Both dresses had lace on the arms, the 1897 dress had &quot;long lace mittens&quot; and the 1895 dress had &quot;long ruffles of old lace.&quot;
* The headgear is not identical but could easily and inexpensively have been changed between the two balls.

==== Madame de Polignac ====
Madame de Polignac was Yolande Martine Gabrielle de Polastron (1749–1793), known as Gabrielle.&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|date=2024-10-01|title=Yolande de Polastron|url=https://en.wikipedia.org/wiki/Yolande_de_Polastron|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Yolande_de_Polastron.&lt;/ref&gt; She was a confidante and favorite of Marie Antoinette beginning in 1775, though Gabrielle was unpopular and the queen's affections were variable.&lt;ref name=&quot;:4&quot; /&gt; One of Madame de Polignac's closest friends was Georgiana, Duchess of Devonshire. Gabrielle died of cancer in Switzerland, shortly after Marie Antoinette's execution in Paris, having fled France after the storming of the Bastille.&lt;ref name=&quot;:4&quot; /&gt;

[[File:King Solomon.jpg|thumb|alt=Old painting of a seated, bearded man holding a scepter and looking off to our left|Simeon Solomon's ''King Solomon'', c. 1874]]
=== The Hon. George Keppel ===
George Keppel (who is early in the list '''of the people attending the ball''' for the London ''Morning Post''), walked in the [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#Louis XV and XVI Period|&quot;Oriental&quot; procession]] as King Solomon in the Suite of Men following the two Queens of Sheba (Lady Cynthia Graham and [[Social Victorians/People/Pless |Daisy, Princess of Pless]])&lt;ref name=&quot;:1&quot; /&gt;{{rp|7, Col. 5b}}&lt;ref&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt; and was attended by &quot;Messrs. [[Social Victorians/People/Halifax|Gordon Wood]] and [[Social Victorians/People/Sarah Spencer-Churchill Wilson|Wilfred Wilson]].&quot;&lt;ref name=&quot;:2&quot; /&gt;{{rp|p. 34, Col. 3a}}

No photograph of Keppel in his costume from this ball exists. The c. 1874 portrait of King Solomon (right), by Simeon Solomon, is now in the National Gallery of Art, Washington, D.C., which received it as a gift in 1995 from William B. O'Neal.&lt;ref name=&quot;:5&quot;&gt;&quot;Solomon, Simeon (1874?)), King Solomon.&quot; Art Object Page 76152. National Gallery of Art. https://www.nga.gov/collection/art-object-page.76152.html#provenance (retrieved 2024-11-27).&lt;/ref&gt; O'Neal may have bought it from the Durlacher Brothers,&lt;ref name=&quot;:5&quot; /&gt; an art gallery founded in London in 1843 by Henry Durlacher and his brother George. After Henry Durlacher's death his sons opened a New York branch.&lt;ref&gt;{{Cite journal|date=2024-05-13|title=Lewis Durlacher|url=https://en.wikipedia.org/wiki/Lewis_Durlacher|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lewis_Durlacher.&lt;/ref&gt; Where George Keppel might have seen this painting — if he ever did — is not clear, because it is not clear how long the Durlacher Brothers owned it or when they exhibited it. In spite of how it looks, this is the entire painting.

Of the many depictions of King Solomon, Simeon Solomon's painting might have been familiar to Keppel or his costumier, and it shows the king wearing a crown, robe and light-colored tunic, like Keppel. 

According to the newspaper reports Keppel was dressed as
*&quot;King Solomon. Tunic of white silk with an elaborate border of jewels; turquoise silk robe lined with white, and a jewelled crown.&quot;&lt;ref name=&quot;:1&quot; /&gt;{{rp|p. 7, Col. 7b}}
*&quot;King Solomon. Tunic of white silk with an elaborate border of jewels; turquoise silk robe, lined with white, and a jewelled crown.&quot;&lt;ref name=&quot;:3&quot;&gt;“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.&lt;/ref&gt;{{rp|p. 3, Col. 2a}}
*(King Solomon), tunic of white silk, embroidered in gold, with an elaborate border on the bottom of jewels and turquoise; turquoise silk robe lined with white; jewelled headdress.&quot;&lt;ref name=&quot;:2&quot;&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032. Print p. 50, Col. 3a.&lt;/ref&gt;{{rp|p. 34, Col. 3a}}

==== King Solomon ====
Stories about King Solomon appear in Jewish, Christian, Islamic and Baháʼí traditions.&lt;ref&gt;{{Cite journal|date=2024-11-04|title=Solomon|url=https://en.wikipedia.org/wiki/Solomon|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Solomon.&lt;/ref&gt; The Queen of Sheba visited King Solomon with gifts and tested his wisdom, perhaps the characteristic most associated with him. The [[Social Victorians/Victorian Things#Encyclopaedia Britannica|9th edition of the ''Encyclopaedia Britannica'']] does not have an article about King Solomon, although he figures in other, historical articles, like the one on Israel.

== The 1895 Warwick Bal Poudré ==
The Hon. George and Alice Keppel attended the Countess and Earl of Warwick's [[Social Victorians/1895 Bal Poudre Warwick Castle|February 1895 bal poudré at Warwick Castle]].

According to the '''''Leamington''' Spa Courier'', Alice Keppel was dressed as a &quot;Lady, time Louis XVI.&quot; at the Warwick ball and wearing a&lt;blockquote&gt;Gown of shell pink satin, pointed bodice, with full paniers, of antique brocade of the real deep rose shade known as du Barri sewn with silver thread and bouquets of roses. Full petticoat, of dull creamy-tinted satin, with a deep band round it of silver tissue embroidered with garlands of small leafless roses. The sleeves had long ruffles of old lace. The hair was powdered and dressed elaborately and high, with three rose du Barri feathers in it and a little cap of lace. The shoes were of pink satin, with diamond buckles.&lt;ref&gt;&quot;The Grand Bal Poudre at Warwick Castle.&quot; ''Leamington Spa Courier'' 09 February 1895, Saturday: 6 [of 8], Cols. 1a–6c [of 6] – 7, Col. 1a. ''British Newspaper Archive''  [https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18950209/042/0006# https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18950209/042/0006].&lt;/ref&gt; (6, 4c)&lt;/blockquote&gt;[[File:Alice - Kostümfest.jpg|thumb|alt=Old black-and-white photograph showing a woman dressed in a long gown with a lot of ruffles|Alice Keppel, 1895?]]Even though no portrait of Alice in her costume from the 1897 Duchess of Devonshire ball exists, a photograph of her in another costume does. This photograph (right), which may come from 1895 but has no provenance information, is a low-resolution digital image of a positive rather than a negative, which explains its poor quality. (The Lafayette Negative Archive [http://lafayette.org.uk/&lt;nowiki&gt;] and the Bassano Studio Portrait Collection [&lt;/nowiki&gt;https://www.npg.org.uk/collections/about/photographs-collection/bassano-studio-portrait-collection&lt;nowiki&gt;] do not include any portraits of the Keppels.)&lt;/nowiki&gt;

The description of Keppel's 1895 costume does not match the dress in this photograph in a number of particulars, at least as far as it is possible to analyze the image. The dress in the photograph is not a dress from the 18th century, and it does not appear to have &quot;[[Social Victorians/Terminology#Hoops|full paniers]],&quot; although it does have a bustle. The late-19th-century neckline, however, could be suggesting an 18th century theatrical costume. The sleeves also have &quot;long ruffles&quot; of lace and, because of the [[Social Victorians/Terminology#Frou-frou|frou-frou]], also seem 18th century.

== Demographics ==
*Nationality: British
=== Residences ===
*1 February 1895: 2, Wilton Crescent, London&lt;ref&gt;&quot;The Grand Bal Poudre at Warwick Castle.&quot; ''Leamington Spa Courier'' 09 February 1895, Saturday: 6 [of 8], Cols. 1a–6c [of 6] – 7, Col. 1a. ''British Newspaper Archive''  [https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18950209/042/0006# https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18950209/042/0006].&lt;/ref&gt; (6, Col. 4c)
*30 Portman Square&lt;ref name=&quot;:0&quot; /&gt;
== Family ==

* William Coutts Keppel, 7th Earl of Albemarle (15 April 1832 – 28 August 1894)&lt;ref&gt;&quot;William Coutts Keppel, 7th Earl of Albemarle.&quot; &quot;Person Page 16514.&quot; ''The Peerage: A Genealogical Survey of the Peerage of Britain as well as the Royal Families of Europe'' https://www.thepeerage.com/p1652.htm#i16514 (accessed November 2022).&lt;/ref&gt;
* Sophia Mary MacNab (5 July 1832 – 5 April 1917)&lt;ref&gt;&quot;Sophia Mary MacNab.&quot; &quot;Person Page 16517.&quot; ''The Peerage: A Genealogical Survey of the Peerage of Britain as well as the Royal Families of Europe'' https://www.thepeerage.com/p1652.htm#i16517 (accessed November 2022).&lt;/ref&gt;

# Lt.-Col. Arnold Allen Cecil Keppel, 8th Earl of Albemarle (1 June 1858 – 12 April 1942)
# Gertrude Mary Keppel (9 November 1859 – 7 April 1860)
# Lady Theodora Keppel (11 January 1862 – 30 October 1945)
# '''Hon. Sir Derek William George Keppel''' (7 April 1863 – 26 April 1944)
# Lady Hilda Mary Keppel (29 August 1864 – 7 October 1955)
# '''Lt.-Col. Hon. George Keppel''' (14 October 1865 – 22 November 1947)
# Lady Leopoldina Olivia Keppel (14 November 1866 – 9 August 1948)
# Lady Susan Mary Keppel (5 May 1868 – 26 June 1953)
# Lady Mary Stuart Keppel (15 May 1869 – 21 September 1906)
# Lady Florence Cecilia Keppel (24 February 1871 – 30 June 1963)


*Freddie (Alice Frederica) Edmonstone Keppel (29 April 1868 – 11 September 1947)&lt;ref name=&quot;:0&quot; /&gt;
*George Keppel (14 October 1865 – 22 November 1947)&lt;ref&gt;{{Cite journal|date=2020-07-14|title=George Keppel (British Army officer, born 1865)|url=https://en.wikipedia.org/w/index.php?title=George_Keppel_(British_Army_officer,_born_1865)&amp;oldid=967698366|journal=Wikipedia|language=en}}&lt;/ref&gt;
#Violet Trefusis (6 June 1894 – 1 March 1970 [Wikipedia says 1972])
#Sonia Cubitt (24 May 1900 – 16 August 1986)

== Notes and Questions ==
# David Cannadine says of courtiers with aristocratic connections and long careers, &quot;Sir Derek Keppel, brother of the eighth Earl of Albermarle, served every sovereign from Queen Victoria to King George VI.&quot;&lt;ref&gt;Cannadine, David. ''The Decline and Fall of the British Aristocracy''. New York: Yale University Press, 1990.&lt;/ref&gt;{{rp|245}}
# The Hon. George Keppel is #39 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who were present]]; the Hon. Mrs. George Keppel — Freddie — is #231.
# Lamont-Brown, Raymond. ''Alice Keppel and Agnes Keyser: Edward VII's Last Loves''. History Press, 2013. Rpt. of ''Edward VII's Last Loves'', Sutton, 2005. Google Books: https://books.google.com/books?id=8LQTDQAAQBAJ.
# When were the Keppels presented to the Queen?
# George Keppel was attended by &quot;Messrs. [[Social Victorians/People/Halifax|Gordon Wood]] and [[Social Victorians/People/Sarah Spencer-Churchill Wilson|Wilfred Wilson]].&quot;

== Footnotes ==
{{reflist}}</text>
      <sha1>am94nnp6miruwwvlnpyhesfr2uya3ak</sha1>
    </revision>
  </page>
  <page>
    <title>User talk:IanVG</title>
    <ns>3</ns>
    <id>274033</id>
    <revision>
      <id>2807801</id>
      <parentid>2802158</parentid>
      <timestamp>2026-05-06T12:47:07Z</timestamp>
      <contributor>
        <username>MediaWiki message delivery</username>
        <id>983498</id>
      </contributor>
      <comment>/* Wikimedians for Sustainable Development - April 2026 Newsletter */ new section</comment>
      <origin>2807801</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="15176" sha1="dngsuu14ag783xwzgopu1f02639xbei" xml:space="preserve">{{Robelbox|theme=9|title=Welcome!|width=100%}}
&lt;div style=&quot;{{Robelbox/pad}}&quot;&gt;
'''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] IanVG!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOUI JS signature icon LTR.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity.

To [[Wikiversity:Introduction|get started]], you may
&lt;!-- The Left column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]].
* Visit a (kind of) [[Wikiversity:Random|random project]].
* [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]].
* Find out about [[Wikiversity:Research|research]] activities on Wikiversity.
* [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left.
&lt;/div&gt;
&lt;!-- The Right column --&gt;
&lt;div style=&quot;width:50.0%; float:left&quot;&gt;
* Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier.
* Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity.
* Give [[Wikiversity:Feedback|feedback]] about your initial observations.
* Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]].
* [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|&lt;kbd&gt;#wikiversity&lt;/kbd&gt;]].
&lt;/div&gt;
&lt;br clear=&quot;both&quot;/&gt;

You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:06, 27 April 2021 (UTC)&lt;/div&gt;
&lt;!-- Template:Welcome --&gt;
{{Robelbox/close}}

== Thank you Dave! ==

Well, I am obviously replying far too late, but thank you @[[User:Dave Braunschweig|Dave Braunschweig]] for your kind words. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 22:25, 29 July 2025 (UTC)

== Possible copyright problem ==
FYI, I just tagged [[Introduction to thermodynamics/Extensive and intensive properties]] for a possible copyright problem: CC NonCommercial not compatible with CC-BY-SA. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:41, 8 November 2025 (UTC)

:Hey @[[User:Dan Polansky|Dan Polansky]]! Thanks a ton, I didn't know that the Non-Commercial part of the license their work was licensed under led to that kind of copyright issue! Thanks for education and help :-). I haven't yet reached out, but I'm going to look for a format email that maybe someone else has already written on the wikipedia to help with this. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:27, 12 November 2025 (UTC)
::Okay, just sent an email it looks something like:
::Dear Dr. Claire Yu Yan,
::I wanted to let you know that I enjoyed your Introduction to Engineering Thermodynamics at https://pressbooks.bccampus.ca/thermo1/, which I found while researching for the free, online and open learning resources on &quot;Wikiversity&quot; (sister project of &quot;Wikipedia&quot;); I thought that your information on the subject is worthy of inclusion in our living and growing document. 
::Wikiversity (
::https://www.wikiversity.org
::) is a Wikimedia Foundation project that is collaboratively edited by volunteers from around the world. Our goal is to create and host a living, and diverse set of learning resources, learning projects and research for use by teachers, students and researchers. It is not only available at no charge, but also freely distributed. It is one of many projects of the non-profit Wikimedia Foundation.
::We can only use your material if you are willing to grant permission for it to be used under terms of the Creative Commons Attribution-ShareAlike 4.0 International License. This means that although you retain the copyright and authorship of your own work, you are granting permission for all others (not just Wikipedia) to use, copy, and share your materials freely—and even potentially use them commercially—as long as they do not try to claim the copyright themselves, or try to prevent others from using or copying them freely. You can read this license in full at:  
::(https://en.wikipedia.org/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License)
::Please note that your contributions may not remain intact as submitted; this license, and the collaborative nature of our project, also entitles others to edit, alter, and update them at will, i.e., to keep up with new information, or suit the text to a different purpose. However, the license also expressly protects authors &quot;from being considered responsible for modifications made by others&quot; while ensuring that those authors get credit for their work. There is more information on our copyright policy at:  
::(https://en.wikipedia.org/wiki/Wikipedia:Copyrights)
::We choose the Creative Commons Attribution-ShareAlike 4.0 International License because it is the best available tool for ensuring that our encyclopedia is and can remain free for all to use, and for providing credit to everyone who donates text and images. It may or may not be compatible with your goals in creating the materials available on your website — that is your choice. Please be assured that if you do not grant permission, your materials will **not** be used at Wikipedia; we have a very strict policy against copyright violations. 
::If you do agree to grant permission, we will credit you for your work in the resulting article's references section, by stating it was based on your work and is used with your permission, and by providing a link back to your website.
::You are obviously an expert in your field, and we invite your active collaboration in writing and editing articles on this subject and any others that might be attractive to you. If you are interested, please see:  
::([[Wikiversity:Introduction|https://en.wikiversity.org/wiki/Wikiversity:Introduction]]) for more information!
::Thank you for your time; we look forward to your reply.
::Kindly,  
::&lt;My name&gt; [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:48, 12 November 2025 (UTC)

== Wikimedians for Sustainable Development - January 2026 Newsletter ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;This is our fifty first newsletter. This issue has news related to SDG 15.&lt;div style=&quot;column-count:2; column-width: 400px;&quot;&gt;

; User group news
*  The [[m:Wikimedians for Sustainable Development/Reports/2025|annual report for 2025]] was published.
* [[m:Wikimedians for Sustainable Development/Next meeting|Next user group meeting]] is 22 February.
* The drafting of the [[m:Wikimedians for Sustainable Development/Annual plan 2026|2026 annual plan]] is under way, please help.

; Other news
* [https://diff.wikimedia.org/2026/01/09/winning-images-of-the-special-category-human-rights-and-environment-from-wiki-loves-earth-2025%F0%9F%A4%9D/ Winning images of the special category “Human Rights and Environment” from Wiki Loves Earth 2025🤝] (SDG 15)

This message was sent with [[m:Special:MyLanguage/Global_message_delivery|Global message delivery]] by &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[m:User:Ainali|Ainali]] ([[m:User talk:Ainali|talk]])&lt;/bdi&gt; 14:22, 4 February 2026 (UTC) •  [[m:Wikimedians for Sustainable Development/Newsletter|Contribute]] • [[m:Global message delivery/Targets/Wikimedians for Sustainable Development newsletter|Manage subscription]]
&lt;/div&gt;
&lt;/div&gt;
&lt;!-- Message sent by User:Ainali@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Wikimedians_for_Sustainable_Development_newsletter&amp;oldid=29928029 --&gt;

== Wikimedians for Sustainable Development - February 2026 Newsletter ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;This is our fifty second newsletter. This issue has news related to SDG 4, 5, 10, 13, 15, 16 and 17.&lt;div style=&quot;column-count:2; column-width: 400px;&quot;&gt;

; User group news
* A proposal for a climate and sustainability meetup at Wikimania has been submitted. Keep your fingers crossed it gets accepted!

; Other news
* [https://metabase.wikibase.cloud Metabase], a project to create a [[m:Movement Strategy/Initiatives/Knowledge Base|movement-wide knowledgebase for activities and initiatives]], now has the property [https://metabase.wikibase.cloud/wiki/Property:P109 relates to sustainable development goal, target or indicator] and all the Sustainable Development Goals, Targets and Indicators. This makes it possible to make sure that your projects and initiative that supports these are marked as doing so and also find previous efforts related to them.
* [https://diff.wikimedia.org/2026/02/11/wiki-for-botanists-why-thematic-engagement-matters/ Wiki for Botanists: Why thematic engagement matters] (SDG 15)
* [https://diff.wikimedia.org/2026/02/15/influence-of-seasonal-and-eco-climatic-factors-on-butterfly-diversity-insights-from-wiki-loves-butterfly/ Influence of Seasonal and Eco-climatic Factors on Butterfly Diversity: Insights from Wiki Loves Butterfly] (SDG 15)
* [https://diff.wikimedia.org/2026/02/15/african-women-in-climate-action-a-continued-editing-journey-through-the-edither-africa-contest-2026/ African Women in Climate Action: A Continued Editing Journey through the EditHer Africa Contest 2026] (SDG 5 &amp; 13)

; Events
* March is Women's History Month and also has the Internaltional Women's day, so there are plenty of related events. Check out [[m:Special:AllEvents|Special:AllEvents]] to find some near you. (SDG 5)
* [[m:Wiki Loves Ramadan 2026|Wiki Loves Ramadan 2026]] (SDG 16)
* [[d:Wikidata:WikiProject_India/Events/International_Mother_Language_Day_2026_Datathon|International Mother Language Day 2026 Datathon]] (SDG 4, 10 &amp;17)

This message was sent with [[m:Special:MyLanguage/Global_message_delivery|Global message delivery]] by &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[m:User:Ainali|Ainali]] ([[m:User talk:Ainali|talk]])&lt;/bdi&gt; 12:13, 2 March 2026 (UTC) •  [[m:Wikimedians for Sustainable Development/Newsletter|Contribute]] • [[m:Global message delivery/Targets/Wikimedians for Sustainable Development newsletter|Manage subscription]]
&lt;/div&gt;
&lt;/div&gt;
&lt;!-- Message sent by User:Ainali@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Wikimedians_for_Sustainable_Development_newsletter&amp;oldid=29928029 --&gt;

== Wikimedians for Sustainable Development - March 2026 Newsletter ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;This is our fifty third newsletter. This issue has news related to SDG 15.&lt;div style=&quot;column-count:2; column-width: 400px;&quot;&gt;

; User group news
* There is now a [[c:Template:User Wikimedians for Sustainable Development|user box template on Wikimedia Commons]] that you can use to show that you are participant of the user group. There were already user boxes on [[m:Template:User Wikimedians for Sustainable Development|Meta]], [[d:Template:User Wikimedians for Sustainable Development|Wikidata]], [[w:en:Template:User Wikimedians for Sustainable Development|English]] and [[w:sv:Mall:Användare Wikimedians for Sustainable Development|Swedish]] Wikipedia. If your home wiki uses user boxes but lacks one, feel free to copy any of these to it.

; Other news
* [https://wikimediafoundation.org/news/2026/03/02/the-winners-of-wiki-loves-earth-2025/ “Cinematic intensity”: The winners of Wiki Loves Earth 2025] (SDG 15)
* [https://www.nature.com/articles/d41586-026-00940-y Scientists should join collaborative online editing communities for biodiversity] (SDG 15)

This message was sent with [[m:Special:MyLanguage/Global_message_delivery|Global message delivery]] by &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[m:User:Ainali|Ainali]] ([[m:User talk:Ainali|talk]])&lt;/bdi&gt; 11:26, 1 April 2026 (UTC) •  [[m:Wikimedians for Sustainable Development/Newsletter|Contribute]] • [[m:Global message delivery/Targets/Wikimedians for Sustainable Development newsletter|Manage subscription]]
&lt;/div&gt;
&lt;/div&gt;
&lt;!-- Message sent by User:Ainali@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Wikimedians_for_Sustainable_Development_newsletter&amp;oldid=30155800 --&gt;

== Wikimedians for Sustainable Development - April 2026 Newsletter ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;This is our fifty fourth newsletter. This issue has news related to SDG 2, 5, 6, 7, 13 and 15.&lt;div style=&quot;column-count:2; column-width: 400px;&quot;&gt;

; News
* [[diffblog:2026/04/15/from-lens-to-knowledge-citizen-science-through-wiki-loves-butterfly/|From Lens to Knowledge: Citizen Science through Wiki Loves Butterfly]] (SDG 15)
* [[outreach:GLAM/Newsletter/March 2026/Contents/Biodiversity Heritage Library report|Wikidata type specimen data model]]  (SDG 15)
* [[outreach:GLAM/Newsletter/March 2026/Contents/Macedonia report|Edit-a-thon &quot;Women Botanists&quot; and “Plants Around Us: Veles” workshop]] (SDG 5 &amp; 15)

; Events
* Ongoing: [[w:en:Wikipedia:100 Days 100 Edits|100 Days 100 Edits]] (SDG 13)
* Ongoing: [[m:Wiki for Sustainable Futures 2026|Wiki for Sustainable Futures 2026]] (SDG 2, 6 &amp; 7)
* May 9-10: [[w:sv:Wikipedia:Projekt naturgeografi/Fotosafari: Fåglar i Skåne 2026|Bird photography trip]] in south Sweden (SDG 15)
* May 30: [[w:sv:Wikipedia:Skrivstuga/Biologisk mångfald|Editathon about biodiversity]] in Stockholm (SDG 15)

This message was sent with [[m:Special:MyLanguage/Global_message_delivery|Global message delivery]] by &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[m:User:Ainali|Ainali]] ([[m:User talk:Ainali|talk]])&lt;/bdi&gt; 12:47, 6 May 2026 (UTC) •  [[m:Wikimedians for Sustainable Development/Newsletter|Contribute]] • [[m:Global message delivery/Targets/Wikimedians for Sustainable Development newsletter|Manage subscription]]
&lt;/div&gt;
&lt;/div&gt;
&lt;!-- Message sent by User:Ainali@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Wikimedians_for_Sustainable_Development_newsletter&amp;oldid=30472224 --&gt;</text>
      <sha1>dngsuu14ag783xwzgopu1f02639xbei</sha1>
    </revision>
  </page>
  <page>
    <title>C language in plain view</title>
    <ns>0</ns>
    <id>285380</id>
    <revision>
      <id>2807812</id>
      <parentid>2807643</parentid>
      <timestamp>2026-05-06T13:41:28Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Applications */</comment>
      <origin>2807812</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="9878" sha1="1ia8xhxg631s2gqy4h65t5w6ls7ovna" xml:space="preserve">=== Introduction ===
* Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]])
* Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]])
* Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]])

=== Handling Repetition ===
* Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]])
* Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]])

=== Handling a Big Work ===
* Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]])
* Functions &amp; Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]])
* Functions &amp; Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]])
* Functions &amp; Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]])

=== Handling Series of Data ===

==== Background ==== 
* Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]])
 
==== Basics ====
* Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]])
* Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]])
* Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]])
* Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]])
* Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]])

==== Examples ==== 
* Spreadsheet Example Programs
:: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]])
:: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]])
:: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]])
:: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]])

==== Applications ==== 
* Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260506.pdf |A.pdf]])
* Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]])
* Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]])
* Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]])
* Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]])
* Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]])
* Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]])

=== Handling Various Kinds of Data ===
* Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]])
* Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]])
* Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]])
* Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]])

=== Handling Low Level Operations ===
* Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]])
* Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]])
* Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]])
* Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]])

=== Declarations ===
* Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]])
* Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]])
* Scope 


=== Class Notes ===

* TOC ([[Media:TOC.20171007.pdf |TOC.pdf]])
* Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library
* Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements
* Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers
* Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts
* Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops
* Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control
* Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions
* Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope
* Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion
* Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions
* Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications
* Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions
* Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications
* Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) 
* Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2)
* Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO
* Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions
* Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications
* Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum
* Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List 
* Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing
* Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing


&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;
See also https://cprogramex.wordpress.com/

== '''Old Materials '''==

until 201201
* Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]])
* Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]])
* Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]])
* Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]])
* Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]])
* Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]])
* Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]])
* Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]])
* Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]])
* Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]])
* Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]])
* Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]])
* Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]])
 
&lt;br&gt;

until 201107
* Intro.1.A ([[Media:Intro.1.A.pdf |pdf]])
* Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]])
* Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]])
* Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]])
* Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]])
* Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]])
* Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]])
* Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]])
* Array.1.A ([[Media:Array.1.A.pdf |pdf]])
* Type.1.A ([[Media:Type.1.A.pdf |pdf]])
* Structure.1.A ([[Media:Structure.1.A.pdf |pdf]])


go to [ [[C programming in plain view]] ]

[[Category:C programming language]]

&lt;/br&gt;</text>
      <sha1>1ia8xhxg631s2gqy4h65t5w6ls7ovna</sha1>
    </revision>
  </page>
  <page>
    <title>Motivation and emotion/Book/2025/Trauma and attachment development</title>
    <ns>0</ns>
    <id>323109</id>
    <revision>
      <id>2807852</id>
      <parentid>2762565</parentid>
      <timestamp>2026-05-07T04:53:02Z</timestamp>
      <contributor>
        <username>CommonsDelinker</username>
        <id>9184</id>
      </contributor>
      <comment>Removing [[:c:File:John_Bowlby_Image.jpg|John_Bowlby_Image.jpg]], it has been deleted from Commons by [[:c:User:Krd|Krd]] because: No license since 29 April 2026.</comment>
      <origin>2807852</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="23857" sha1="7ui47ojkptxvx4kioq5427bg460tezq" xml:space="preserve">{{title|Trauma and attachment development:&lt;br&gt;How does early trauma shape the formation of attachment styles?}}
__TOC__

==Overview==
{{RoundBoxTop|theme=4}}
[[File:Couple arguing.png|thumb|160x160px|'''Figure 1.''' Insecure attachment causes friction in a relationship]]

; They love me, they love me not?

Shay and Oliver have been seeing each other for two months. So far, everything has been going smoothly. Until one day{{g}}, Oliver tells Shay he loves her and wants to make it official. Shay says yes but Oliver notices she has become distant. It seems the more he tries to get close to her, the more she pulls away. He tries to bring it up with her, but it seems to frustrate her even more.

Oliver feels rejected and hurt. Should he accept that maybe their relationship just doesn't work?
{{RoundBoxBottom}}

Shay and Oliver’s story reflects a dynamic familiar to many couples: one partner becomes the peacemaker, often sacrificing their own emotional needs to maintain harmony, while the other remains emotionally distant or unpredictable. This pattern often leads to unresolved tension, dissatisfaction, or eventual breakdown in the relationship.

Rather than focusing solely on the surface-level issues in adult relationships, psychological research increasingly{{f}} points to early life experiences as a root cause. Studies{{f}} in attachment theory suggest that the emotional bonds formed with primary caregivers during early childhood can have a lasting impact on how individuals relate to romantic partners later in life. These influential interactions contribute to the development of secure or insecure attachment styles, which in turn influence communication, trust, and emotional regulation in adult relationships{{f}}.

This chapter explores how early attachment experiences, particularly those shaped by trauma, can influence adult romantic relationships. How do different attachment styles manifest in adulthood? And can psychology aid in changing insecure attachment patterns{{g}}.

{{RoundBoxTop|theme=4}}
'''Focus questions'''

*What is attachment theory?
*What are the four different attachment styles?
*What is my attachment style?
*How does trauma play a role{{ic|in what}}?
*Can we become more secure?{{ic|Use open-ended questions}}

{{RoundBoxBottom}}

== What is attachment theory? ==

[[wikipedia:Attachment_theory|Attachment theory]], developed by [[wikipedia:John_Bowlby|John Bowlby]] (see Figure 2), emerged as an alternative to [[wikipedia:Psychoanalytic_theory|psychoanalytic theory]], aiming to explain why separation from caregivers causes anxiety in children (Bolen, 2000). Bowlby proposed that there is a critical period during early childhood (around 2.5 years old) when children must form secure attachments to their primary caregivers for healthy emotional and social development (McLeod, 2025). 

These early bonds can be either secure or insecure, and they play a crucial role in shaping how individuals interact in relationships throughout their lives{{f}}. Attachment security has been consistently linked to better social and emotional adjustment across the lifespan (Doyle et al., 2000). Securely attached children tend to develop stronger emotional regulation and healthier social connections, while those with insecure attachment styles may struggle with intimacy and emotional balance later on{{f}}.

One of the primary methods used to assess attachment in children is the [[wikipedia:Strange_situation|Strange Situation Procedure]] (SSP), developed by [[wikipedia:Mary_Ainsworth|Mary Ainsworth]]. This procedure involves a series of brief separations and reunions with the caregiver, designed to activate the child’s attachment system and reveal their attachment style (Ainsworth et al., 1978). It has become a cornerstone{{f}} in understanding how early attachment influences behaviour in later relationships.

=== What are the different attachment styles? ===
Within attachment theory {{g}} there are a four different subtypes or styles of attachment (see Table 1).

==== Different attachment styles ====
{{ic|Add APA style table caption with citation(s) to sources of this information}}
{| class=&quot;wikitable&quot;
!Attachment 
!Caregiver Behaviours
!Characteristics
|-
|Secure
|emotionally available and responsive
to child's needs
|can form strong, intimate connections, communicate well, and manage stress and conflict
|-
|Anxious-Avoidant
|neglectful or emotionally unavailable
|uncomfortable with intimacy, can be emotionally distant and prefers to rely on themselves rather than others for emotional support.
|-
|Disorganized
|inconsistent or frightening 
|erratic and unpredictable behaviour, can be contradictory in actions
|-
|Anxious 
|inconsistent or neglectful caregiving
|high need for reassurance and fear of abandonment. Can be overly dependent on partner for validation
|}
'''Anxious-Avoidant Attachment''' 

Anxious-avoidant attachment sometimes known as dismissive avoidant attachment usually develops from a caregiver who was neglectful or emotionally unavailable. People with this attachment style tend to feel uncomfortable with intimacy and prefer to rely on themselves rather than others for emotional support.{{f}}

'''Secure Attachment'''

Secure attachment happens when a caregiver is emotionally available and responsive to the child’s needs. Individuals with secure attachment can form strong, intimate connections, communicate well, and manage stress and conflict better than those with insecure attachment styles. They’re comfortable with both closeness and independence in relationships.{{f}}

'''Disorganized Attachment'''

Disorganized attachment also known as fearful avoidant often develops when a caregiver is inconsistent or frightening. People with this attachment style show erratic and unpredictable behaviour, sometimes seeking closeness but other times pushing people away. Their actions can appear contradictory, leaving them feeling confused and unsure in relationships.{{f}}

'''Anxious Attachment'''

Anxious attachment usually develops from inconsistent or neglectful caregiving. People with this attachment style are marked by a high need for reassurance, fear of abandonment, clinginess, and jealousy. They can become overly dependent on their partner for validation and tend to get caught up in emotional highs and lows.{{f}}

&lt;quiz display=&quot;simple&quot;&gt;
{Anxious avoidant attachment is a secure attachment style{{ic|the answer is incorrect}}:
|type=&quot;()&quot;}
+ True
- False

{Other non care-giver related traumas can lead to insecure attachment{{ic|the answer is incorrect}}:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

=== Limits of attachment theory ===
Though{{sp}} attachment theory is widely used around the world{{f}}, it faces several important critiques{{f}}. Firstly, it is quite Western-centric, often reflecting parenting styles common in Western cultures, such as frequent attention and constant availability from caregiver, which may not apply across cultures{{f}}. This limits the theory’s generalisability across different cultural contexts (Rothbaum et al., 2000; Rothbaum et al., 2001). It also places a heavy emphasis on maternal influence, which can overlook the important roles of fathers and other caregivers (Lamb &amp; Bornstein, 2013). Lastly, attachment theory tends to downplay other key influences like genetics, socioeconomic background, and temperament, which can result in an oversimplified view of child development (Bronfenbrenner, 1979). However, despite these criticisms, attachment theory remains a valuable framework for understanding early relationships and emotional development. Its core ideas have been expanded and adapted over time, and newer research{{f}} continues to build on its foundations while addressing its cultural and contextual limitations.

== Insecure attachment and relationship instability ==
Insecure attachment can be categorized into avoidant, anxious, and disorganised attachment styles. Unlike securely attached individuals, those with insecure attachment often develop in environments where caregivers are inconsistent, unavailable, or unresponsive during times of distress{{f}}. As a result, they may adopt certain strategies such as avoidance or hyperactivation to cope with these needs.{{g}}(Dagan et al., 2021) Consequently, children may develop a simultaneous yearning for closeness and a fear of abandonment, depending on the subtype this will often lead to increased sensitivity to others’ emotional states and communication, patterns of excessive rumination, emotional distancing, discomfort with intimacy and contradictory behaviour{{f}}.

[[File:ATT.STYLES.png|thumb|405x405px|'''Figure 3'''. Attachment styles ]]

Insecure attachment has been widely linked to increased vulnerability to internalizing symptoms such as anxiety and depression (Dagan et al., 2021). Bowlby’s foundational work emphasized that early experiences with caregivers, particularly during moments of stress, shape an individual. Particularly expectations, beliefs, and attitudes about themselves and others in close relationships (Simpson &amp; Rholes, 2017). Over time, these models influence how individuals engage in future relationships, especially in how they seek comfort, express needs, or respond to perceived threats of rejection or abandonment.

== How does trauma play a role? ==
Trauma that is experienced in early childhood is foundational to the development of attachment styles{{f}}. Exposure to any variety of trauma for example, severe stress, neglect or abuse rewires the brain’s sense of safety that secure attachment relies on, resulting in insecure or disorganised patterns of relating to others (Lahousen et al., 2019). Specifically, disorganised attachment, has been strongly linked to traumatic caregiving environments where the caregiver is simultaneously a source of fear and comfort (Greenman et al., 2024). This creates an internal conflict in the child, who seeks closeness but also anticipates harm, leading to behaviours led by confusion, fear, and emotional dysregulation. Over time, this can manifest in adulthood as relationship complications, trust issues, and inconsistent behaviour{{f}}. Trauma can also give rise to avoidant or dismissing attachment, in which individuals adopt emotional distancing as a defensive mechanism.{{f}} These individuals tend to suppress emotional needs and downplay the importance of close relationships as a way to avoid further psychological pain (Greenman et al., 2024). Ultimately, early trauma not only shapes how individuals respond to attachment cues but also influences their core beliefs about safety, trust, and intimacy, often carrying these patterns into adulthood unless addressed through supportive relationships or psychological intervention{{f}}.

=== Neurobiological mechanisms of trauma and attachment ===
[[File:Brain regions involved in memory formation.jpg|thumb|405x405px|'''Figure 4'''. The traumatised brain]]

Early trauma can profoundly alter brain development and disrupt the formation of stable cognitive and emotional patterns{{f}}. Rather than maturing in a secure and nurturing environment, the brain adapts in ways that prioritize survival over healthy typical functioning. According to Lahousen, Unterrainer, and Kapfhammer (2019), traumatic or neglectful caregiving during early development interferes with the brain’s stress regulation systems, particularly the [[wikipedia:Hypothalamic–pituitary–adrenal_axis|hypothalamic-pituitary-adrenal (HPA) axis]], resulting in chronic overactivation and elevated [[wikipedia:Cortisol|cortisol]] levels. Prolonged exposure to this heightened stress state can impair key brain structures: the [[wikipedia:Amygdala|amygdala]] (see figure 4) becomes hyperresponsive, increasing fear and threat sensitivity, while the [[wikipedia:Prefrontal_cortex|prefrontal cortex]] which is essential for emotion regulation, empathy, and executive function shows reduced efficiency. Trauma also disrupts the [[wikipedia:Oxytocin|oxytocin]] system by reducing oxytocin release and impairing receptor sensitivity, weakening the brain’s capacity for social bonding, trust, and feelings of safety{{f}}. These neurobiological changes set the foundation for the development of insecure or disorganised attachment styles{{f}}. Instead of fostering secure emotional bonds, the traumatised individual’s brain becomes concentrated on survival, vigilance, emotional dysregulation and self-defence, which can significantly affect the development of healthy interpersonal relationships later in life{{f}}.

=== Non caregiver-specific trauma ===
While early relationships with caregivers play a central role in shaping attachment styles, insecure attachment can also develop later in life as a result of non caregiver-specific trauma{{f}}. Traumatic experiences such as abuse, neglect, loss, or betrayal in romantic relationships, friendships, or even during major life events can disrupt a person's sense of emotional security{{f}}. These later-life traumas can reinforce or even create patterns of insecure attachment. Especially in individuals who may have previously had secure foundations{{g}}. Ogle et al. (2015) found that the relationship between PTSD symptoms and attachment anxiety was stronger in individuals whose trauma occurred in early life, but those with trauma in adulthood were also affected, indicating that attachment insecurity is not limited to childhood experiences. Additionally, attachment anxiety is associated with a tendency to use hyper-activating coping strategies, amplifying negative emotions, overreacting to stress, and perceiving situations as more threatening than they may be (Maunder et al., 2006){{ic|Not in References}}. These strategies can become more pronounced after relational or situational trauma, leading to greater emotional instability and difficulty trusting others{{f}}. This highlights that insecure attachment styles can emerge not only from early caregiving environments but also from the complex experiences and trauma individuals may encounter throughout life.

&lt;quiz display=&quot;simple&quot;&gt;
{Attachment styles were developed by John Bowlby:
|type=&quot;()&quot;}
+ True
- False

{Other non care-giver related traumas can lead to insecure attachment{{ic|Repeated question; incorrect answer}}:
|type=&quot;()&quot;}
- True
+ False
&lt;/quiz&gt;

== From survival to security ==
Although trauma can have a significant and lasting influence on our brains, attachment styles developed by trauma are not completely fixed. Through self-awareness, supportive relationships, and often psychological intervention, individuals can gradually shift from insecure patterns to a more secure attachment style{{f}}. Moving from survival-based coping mechanisms, such as emotional withdrawal, fear of abandonment, or hypervigilance, to a place of emotional safety and trust is possible over time. According to Vrtička and Vuilleumier (2012), developing a positive model of others, alongside positive self-beliefs such as being worthy, lovable, and capable, is key to forming secure attachments. These internal shifts help reframe how individuals perceive relationships, allowing for deeper emotional connections, greater self-regulation, and healthier responses to stressful situations. With these interventions, even those affected by insecure or disorganized attachment can work toward lasting emotional security.

[[File:Emotion-Focused Therapy Illustration.jpg|thumb|330x330px|'''Figure 5'''. Emotion focused therapy]]

=== Rewiring your brain ===
One way individuals with insecure attachment can become securely attached through targeted psychological interventions, especially when individuals become aware of how their early or later-life experiences have shaped their relational patterns{{g}}. Two well-supported approaches are [[wikipedia:Cognitive_behavioral_therapy|Cognitive Behavioural Therapy (CBT)]] and [[wikipedia:Emotionally_focused_therapy|Emotionally Focused Therapy (EFT).]]

==== Cognitive Behavioural Therapy ====
CBT helps individuals explore and challenge the negative core beliefs they may have developed about themselves and others, often rooted in earlier attachment wounds{{f}}. For those with insecure attachment, these beliefs might include “I am unlovable” or “my significant other will hurt or abandon me.” CBT works to restructure these thoughts and encourages the development of healthier emotional responses and interpersonal behaviours{{f}}. According to Cicchetti, Rogosch, and Toth (2006), interventions grounded in developmental science can effectively increase attachment security, particularly in high-risk or maltreated populations. They found that “children in the intervention groups demonstrated substantial increases in secure attachment,” showing that targeted psychological support can shift attachment patterns (p. 638).

==== Emotionally Focused Therapy ====
EFT, developed by Sue Johnson, focuses on reshaping emotional bonds in relationships, particularly by helping individuals identify and express their attachment needs in a safe, structured environment. EFT is especially effective in couples and family therapy, as it works directly with patterns of emotional disconnection and insecurity. Johnson (2019) explains that the goal of EFT is to help individuals and couples move from “desperate, rigid reactive positions” to ones that allow emotional accessibility and responsiveness, hallmarks of secure attachment. By creating new emotional experiences within the therapy setting, EFT allows clients to internalize new models of relating, rooted in safety and trust.

==Conclusion==
The relational dynamic between Shay and Oliver reflects a common issue found in many relationships worldwide. Often one individual assuming the role of emotional caretaker while the other maintains emotional distance or inconsistency. Rather than originating solely from interpersonal conflict, such patterns often have deeper psychological roots in early attachment experiences. Attachment theory suggests that the emotional bonds formed with primary caregivers in childhood shape internal models of self and others, which continue to influence our behaviours, emotional regulation, and desire for intimacy throughout the lifespan. 

This chapter examined how attachment styles, secure, anxious, avoidant, and disorganised, are shaped not only by early caregiving environments but also by significant non-caregiver-specific traumas encountered later in life. Whether rooted in early neglect or later experiences of betrayal, trauma can dysregulate emotional processing and relationship functioning. Neurobiological research has further demonstrated that early trauma affects brain structures responsible for stress regulation and bonding, contributing to the development of insecure attachment styles.

Despite the enduring effects of insecure attachment, these patterns are not fixed. With psychological insight, supportive relationships, and evidence-based interventions, individuals can begin to transition from survival-driven behaviours toward more secure relational patterns. Cognitive Behavioural Therapy helps to restructure maladaptive beliefs and emotional responses, while Emotionally Focused Therapy fosters emotional accessibility and responsiveness in relationships. These therapeutic approaches offer pathways for individuals to repair internal working models and develop a more secure sense of self in relation to others.

While early attachment experiences set a foundation for relationship tendencies, they do not dictate the outcomes of relationships. The potential for emotional growth, healing, and secure attachment remains accessible throughout life, particularly in becoming aware of how trauma can affect the way you relate to other people, psychological interventions and support.

== References ==

{{Hanging indent|1=
Bolen, R. M. (2000). Validity of attachment theory. ''Trauma, Violence &amp; Abuse, 1''(2), 128–153. https://www.jstor.org/stable/26636245  

Bronfenbrenner, U. (1979). ''The ecology of human development: Experiments by nature and design''. Harvard University Press.  

Cicchetti, D., Rogosch, F. A., &amp; Toth, S. L. (2006). Fostering secure attachment in infants in maltreating families through preventive interventions. ''Development and Psychopathology, 18''(3), 623–649. https://doi.org/10.1017/S0954579406060329  

Dagan, O., Groh, A. M., Madigan, S., &amp; Bernard, K. (2021). A lifespan development theory of insecure attachment and internalizing symptoms: Integrating meta-analytic evidence via a testable evolutionary mis/match hypothesis. ''Brain Sciences, 11''(9), 1226. https://doi.org/10.3390/brainsci11091226  

Dargie, E. (2020). Attachment theory in practice: Emotionally focused therapy (EFT) with individuals, couples, and families. ''Journal of Sex &amp; Marital Therapy, 46''(7), 717–719. https://doi.org/10.1080/0092623x.2020.1794183  

Doyle, A. B., Markiewicz, D., Brendgen, M., Lieberman, M., &amp; Voss, K. (2000). Child attachment security and self-concept: Associations with mother and father attachment style and marital quality. ''Merrill-Palmer Quarterly, 46''(3), 514–539. https://www.jstor.org/stable/23093743  

Greenman, P. S., Renzi, A., Monaco, S., Luciani, F., &amp; Di Trani, M. (2024). How does trauma make you sick? The role of attachment in explaining somatic symptoms of survivors of childhood trauma. ''Healthcare, 12''(2), 203. https://doi.org/10.3390/healthcare12020203  

Lahousen, T., Unterrainer, H. F., &amp; Kapfhammer, H.-P. (2019). Psychobiology of attachment and trauma—Some general remarks from a clinical perspective. ''Frontiers in Psychiatry, 10'', 914. https://doi.org/10.3389/fpsyt.2019.00914  

Lamb, M. E. (2013). ''Social and personality development''. Psychology Press. https://doi.org/10.4324/9780203813386  

Levine, A., &amp; Heller, R. (2010). ''Attached: The new science of adult attachment and how it can help you find—and keep—love''. TarcherPerigee.  

McLeod, S. (2025, April 20). John Bowlby’s attachment theory. Simply Psychology. https://www.simplypsychology.org/bowlby.html  

Ogle, C. M., Rubin, D. C., &amp; Siegler, I. C. (2015). The relation between insecure attachment and posttraumatic stress: Early life versus adulthood traumas. ''Psychological Trauma: Theory, Research, Practice, and Policy, 7''(4), 324–332. https://doi.org/10.1037/tra0000015  

Rothbaum, F., Weisz, J., Pott, M., Miyake, K., &amp; Morelli, G. (2000). Attachment and culture: Security in the United States and Japan. ''American Psychologist, 55''(10), 1093–1104. https://doi.org/10.1037/0003-066X.55.10.1093  

Rothbaum, F., Weisz, J., Pott, M., Miyake, K., &amp; Morelli, G. (2001). Deeper into attachment and culture. ''American Psychologist, 56''(10), 827–829. https://doi.org/10.1037/0003-066X.56.10.827  

Simpson, J. A., &amp; Rholes, W. S. (2017). Adult attachment, stress, and romantic relationships. ''Current Opinion in Psychology, 13'', 19–24. https://doi.org/10.1016/j.copsyc.2016.04.006  

Vrtička, P., &amp; Vuilleumier, P. (2012). Neuroscience of human social interactions and adult attachment style. ''Frontiers in Human Neuroscience, 6'', 212. https://doi.org/10.3389/fnhum.2012.00212  
}}

== External links ==
[https://www.amazon.com.au/Attached-Science-Adult-Attachment-YouFind/dp/1585429139 Attached: the new science of adult attachment and how it can help you find - and keep - love] (Amazon.com)

[https://www.goodreads.com/book/show/16158307-insecure-in-love Insecure in love: how anxious attachment can make you feel jealous, needy, and worried and what you can do about it] (goodreads.com)

[https://www.youtube.com/watch?v=QTsewNrHUHU The strange situation - mary ainsworth] (youtube)

[[Category:{{#titleparts:{{PAGENAME}}|3}}]]
[[Category:Motivation and emotion/Book/Attachment]]
[[Category:Motivation and emotion/Book/Trauma]]</text>
      <sha1>7ui47ojkptxvx4kioq5427bg460tezq</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity talk:Artificial intelligence</title>
    <ns>5</ns>
    <id>324777</id>
    <revision>
      <id>2807817</id>
      <parentid>2807629</parentid>
      <timestamp>2026-05-06T14:46:21Z</timestamp>
      <contributor>
        <username>Lbeaumont</username>
        <id>278565</id>
      </contributor>
      <comment>/* Burden of Proof */ new section</comment>
      <origin>2807817</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="44077" sha1="embtk7n26up267jq5rwxa6t924160uv" xml:space="preserve">== My POV ==

*I would remove &quot;The contributor should be an expert on the topic&quot;, because Wikiversity is not about authorities and we are not able to check weather certain person contributing LLM-created text is an expert or not.
*This is not applicable to all situations, when using LLM: &quot;where citations are included.&quot; Sometimes you generate wery short overviews or general things.
[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:41, 16 October 2025 (UTC)
: In my notes, I have a proposal to restrict the use of GenAI even more; it is much more of a threat than an opportunity for the English Wikiversity.
: In the mean time, requiring that a contributor be an expert or at least know what he is writing about is a very good thing, from my perspective. It is not true that we have no way of tentatively determining whether someone is an expert or not: we can ask for self-disclosure and we can test knowledge. And he who does not want to be tested should not be inserting GenAI into mainspace. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 07:44, 3 November 2025 (UTC)
::But the obsession with expertise closes down an open Wikiversity. Nupedia was expert and failed, Wikipedia was open and succeeded. Why should Wikiversity go the way of Nupedia? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 16:13, 17 November 2025 (UTC)
::I agree with Juandev here, although in theory the content added by folks on Wikiversity should be coming from a place of expertise, I also understood one of the missions of wikiversity to be a place where expertise can be actively developed through the act of editing by editors. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:11, 23 March 2026 (UTC)
:::Could this be like that we are letting people make their own mistakes, learn from their own mistakes(if they can) and then only intervene if the project becomes dormant/'paused' or if the contributor is asking for help from us for an extended period of time? Otherwise if we didn't let the users/contributors make mistakes they might not learn from them.
:::I feel like that is how my personal experience has been so far. I got help when I asked for it, otherwise I'm free to develop my draft as well as I can while making mistakes and hopefully learning from them. I guess exceptions for 'intervention' is when a user publishes content with red flags, and I don't know if my mention of Taylor Swift is one but I also need to learn sooner or later who to contact and for what reason. I feel like I'm having a Wikiversity journey but also a personal journey at the same time with and without &quot;AI Mode&quot; by Google and other providers. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 08:24, 2 May 2026 (UTC)
:I think generative AI should be used as a tool. If you are copying the text word for word, the text might not be correct. If you are interested in a particular topic, feel free to use GenAI, but maybe check the facts before using it and provide the link to the conversation. —[[User:RailwayEnthusiast2025|&lt;span style=&quot;font-family:Verdana; color:#008000; text-shadow:gray 0.2em 0.2em 0.4em;&quot;&gt;RailwayEnthusiast2025&lt;/span&gt;]] &lt;sup&gt;[[User talk:RailwayEnthusiast2025|&lt;span style=&quot;color:#59a53f&quot;&gt;''talk with me!''&lt;/span&gt;]]&lt;/sup&gt; 11:44, 3 November 2025 (UTC)
::In that case, some kind of scale should be introduced that a human editor would use to indicate how much LLM was used. From full text created entirely by a chatbot (which I don't think is a good idea, because it may contain errors in the form of hallucinations and at the same time takes away the authorship from the given LLM). To text proofreading and only minor interventions by artificial intelligence. @[[User:RailwayEnthusiast2025|RailwayEnthusiast2025]] [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:36, 1 February 2026 (UTC)

== A proposed caveat on when they are used ==

If we allow generative AI usage, I think we should require disclosure of what tool was used, when, and which prompt(s) it was given. Understanding not only that it was used but how is crucial, plus, since these tools change rapidly, knowing the time/date is also key for understanding what it was likely processing and how when it generated the output. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:47, 8 November 2025 (UTC)

:Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 16:14, 17 November 2025 (UTC)
:I agree that knowing how users are using AI may be a good data for Wikiversity community to learn how AI is used, but I would not overcomplicate the policy. So what about to start this with optional values for {{tl|AI-generated}} tempate? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:41, 23 February 2026 (UTC)

== Different uses of AI ==

I am just pointing out that AI is not just used to generate text, which could be copy paste to Wikiversity. One may use AI to improve their grammar (for example with the use of Grammarly), other one may use GPT to create wiki tabs from CSV. So if the proposed policy is using wide title Artifical inteligence, I would consider all use cases and decide how to deal with them. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 14:38, 22 November 2025 (UTC)

: I believe that if the changes made by an AI in the authoring process falls into the definition of a &quot;minor edit&quot; (borrowing the definitions of [[:w:WP:MINOR]]), the resulting content should not count as &quot;AI-generated&quot;. So definitely not for the table thing. Grammar... depends on how extensive the change is (possibly because I dislike Grammarly; I may be quite biased here). --[[User:Artoria2e5|Artoria2e5]] ([[User talk:Artoria2e5|discuss]] • [[Special:Contributions/Artoria2e5|contribs]]) 05:33, 16 April 2026 (UTC)

== [[Wikiversity:Colloquium#Template:AI-generated]] ==

Discussion on indication of a resource being AI-generated. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 15:55, 26 January 2026 (UTC)

== Confirm AI use is okay ==

Before I continue adding to the the Law School 101 course I started, I want to make sure that it's consistent with the AI policy. I'm seeing some conflicting opinions here that may not be as nuanced as they should be. 

I would not have decided to share the Law School 101 course from an LLM if I didn't feel it was uber good, completely missing in public access, and sorely needed to be available to the public.

I am 100% ok with having an AI disclaimer on the front page of the course, but I'm not going to go and add it to each page with the prompt on each page. That's stupid. Some prompts were &quot;Next class&quot;.

If I went through the course, I'm an expert on the topic of the course. Seriously, though, expertise is an extremely stretchable concept that cannot be used as a whip to disqualify great courses. A person may have had years of education, high IQ, for example. And the topic itself may be at the level of general knowledge where the value of expertise on the topic may much less relevant to the quality of material that the course creator sees in the content. 

And we're moving away from an era when LLMs were producing errors. Of course, all content from an LLM must be vetted, and of course expert opinions on class content are welcome, but to preclude excellent course content from being made public would detract from the mission of Wikiversity.  [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 07:11, 17 February 2026 (UTC)

== Evolving a Wikiversity policy on AI ==

=== Adapting to New Technologies ===
I am old enough to have obtained my BSEE degree in 1972, before the general availability of pocket electronic calculators. I laboriously used a slide rule and pencil and paper for those hundreds of calculations. Since then, I have witnessed the introduction of pocket calculators, scientific calculators, cassette recorders, video recorders, CDs, DVDs, personal computers, spreadsheets, word processors, spell checkers, online dictionaries and thesauruses, cell phones, GPS, the Internet, search engines, grammar checkers, Nanny cams, cloud storage, Napster, streaming, smart phones, Wolfram Alpha, homework assistants, tablets, Wikimedia projects, MOOCs, videoconferencing, Crypto currency, and most recently AI large language models.

Each of these technologies has required us to adapt. We had to be clear about our needs and goals. These goals might include learning, teaching, getting the right answer, efficiency, profit, ease of use, entertainment, sharing, collaboration, safety, intellectual property rights, and no doubt other concerns. 

Technology is inherently morally neutral. A hammer can be used to build a house or to bludgeon someone. How we decide to use technology is our choice, not the destiny of the technology. 

=== Guiding Principles and Lessons Learned ===
It is wise to avoid overreacting or underreacting. 

It is wise to avoid “[[wikipedia:One-drop_rule|one drop rules]]” that indiscriminately, and unnecessarily, prejudice the use of emerging technologies.

It is wise to avoid any form of “[[wikipedia:Satanic_panic|satanic panic]]” that causes unwarranted panic, anxiety, unfounded accusations, and an unfounded search for the guilty. Furthermore, unduly highlighting the use of AI within Wikiversity is a form of [[wikipedia:Ad_hominem|Ad hominem]] attack—attacking the source rather than the argument or resulting text. Doing so pejoratively stains the material, and the authors, with a form of [[wikipedia:The_Scarlet_Letter|scarlet letter]]. 

It is useful to understand and acknowledge the nuances of the many ways that the new technology can be used. Existing LLM’s can be used to:

1)     Proofread copy,

2)     As a thesaurus or to suggest a variety of word choices,

3)     To extend a list of items sharing various characteristics,

4)     To assist in brainstorming, 

5)     To write introductory, summary, or clarifying text.

6)     To suggest alternative wording or rewriting text,

7)     To modify the tone of the text,

8)     To generate a list of questions,

9)     As a research tool to identify likely sources of new information,

10)  To demonstrate the limits and capabilities of the technology, and

11)  in many more ways. 

These are very different uses of the technology, and it is misleading to place them into a single category.

=== Addressing Wikiversity goals. ===
Wikiversity provides “learning resources” freely available to the users. Editors have a responsibility to follow established [[Wikiversity:Policies|Wikiversity Policies]].  

Content [[Wikiversity:Verifiability|must be verifiable]]. While professors have the liberty to profess, ''accurate propositional statements'' typically provide more useful learning resources than do false or misleading propositional statements. As described above, text generated or assisted by an LLM often does not include propositional statements subject to verification. Both people and LLMs sometimes hallucinate (and bloviate) and are otherwise fallible, and therefore what is relevant is the ''accuracy of the propositional statements'', regardless of the source. 

If the editor takes sufficient care and has the expertise to verify the accuracy of the propositional statements made, the origin of those statements is irrelevant, as long as they are properly cited.

Because the source of verified and accurate propositional statements is irrelevant, marking, and especially obtrusive or pejorative marking, of AI generated content is unnecessary. 

Because I recognize that there may be good reasons to collect AI generated materials into a category, I recommend the “AI Generated” template be redesigned to be similar to the “[[:Category:Essays|Essay” category tag]]. This would be a small tag appearing along the right-hand margin of the page. The tag could usefully include a parameter identify the mode of the AI used, as suggested above. 

I hope these ideas are carefully considered as we continue to collaborate in adapting to this new and valuable technology. I also call for a moratorium on defacing existing materials until a consensus policy is adopted. [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 22:45, 10 March 2026 (UTC)

:Thanks for your ideas. I didn't realize this was a draft policy discussion. In this case, please take a look at this AI-generated and human-vetted course &quot;[[Law School 101]].&quot; It is so superb. I have taken it in its entirety, and I believe it's a top-notch learning resource for every adult. It's also pure joy to go through and sets the bar high.
:And I don't see anything online that would accomplish something remotely similar. 95% of undergraduates graduate having no clue what Law is all about, all while it affects every facet of our lives every day. 
:I think this should be a class in colleges, and the Intro part should even be offered in high schools (imagine the thrill of going through the entire one year of law school in ten classes?). I think it's the biggest, sorest gap in core education these days, and it's unclear why the legal professionals are MIA and not scrambling to fill this screaming void.
:Specifically regarding AI use - this debate must not be out of context. And the context is that access to education must not be stifled and veiled behind arbitrary exclusionary barriers.  [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 10:19, 12 March 2026 (UTC)
::I just note that this policy draft is not against AI generated content @[[User:Berkeleywho|Berkeleywho]]. Thats why nobode disputed your previous post and your reflection was build in to the proposal. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:52, 12 March 2026 (UTC)
:::Cool. I understand this is an extremely complex topic on many levels.  [[User:Berkeleywho|Berkeleywho]] ([[User talk:Berkeleywho|discuss]] • [[Special:Contributions/Berkeleywho|contribs]]) 10:56, 12 March 2026 (UTC)
:You said &quot;If the editor takes sufficient care&quot;. But some editors does not take sufficient care. Some editor say its not a policy I dont mind. Thats why this policy is proposed that everbody do that and co-create quality resource on Wikiversity.
:You talk about some embarrassment that a source is marked as LLM-generated, but this rule requires you to mark it yourself and if you don't mark it, we can only suggest it to you. So why rebel against such a practice? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:50, 12 March 2026 (UTC)

== Toward a Justified and Parsimonious AI Policy ==

As we collaborate to develop a consensus policy on the use of Large Language Models, it is wise to begin by considering the needs of the various stakeholders to the policy.

The stakeholders are:

# The users,
# The source providers, and
# The editors

There may also be others with a minor stake in this policy, including the population at large.

The many needs of the users are currently addressed by long-standing [[Wikiversity:Policies|Wikiversity policies]], so we can focus on what, if any, additional needs arise as LLMs are deployed. 

As always, users need assurance that propositional statements are accurate. This is covered by the existing policy on [[Wikiversity:Verifiability|verifiably]]. In addition, it is expected by both the users and those that provide materials used as sources for the text are [[Wikiversity:Cite sources|accurately attributed]]. This is also covered by [[Wikiversity:Cite sources|existing policies]].

To respect the time and effort of editors, a parsimonious policy will unburden editors from costly requirements that exceed benefits to the users.

Finally, it is important to recognize that because attention is our most valuable seizing attention unnecessarily is a form of theft. 

The following proposed policy statement results from these considerations:

=== Recommended Policy statement: ===

* Editors [[Wikiversity:Verifiability|verify the accuracy]] of propositional statements, regardless of the source.
* Editors [[Wikiversity:Cite sources|attribute the source]] of propositional statements. In the case of LLM, cite the LLM model and the prompt used.
* Use of various available templates to mark the use of LLM are optional. Templates that are flexible in noting the type and extend of LLM usage are preferred. Templates that avoid unduly distracting or alarming the user are preferred.

[[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 19:58, 19 March 2026 (UTC)

:Just note, that [[Wikiversity:Cite sources]] is not a policy. You can read it on the top of the page, its a ''proposed policy''. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:28, 19 March 2026 (UTC)
:I would say, that the actual text is better then your first two proposed statements, because:
:#Your proposal is less clear to me, so it might be less clear to others - we need policies which are easy to understand. For example, the course structure generated in LLM is not, in my opinion, a ''propositional statement'', but the rule should still cover such a case.
:#Your proposal is missing the option, when references are outputed by the LLM
:Templates that indicate AI-generated content should be mandatory, as they allow you to create statistics about AI-generated content. This is good for creating tools or other policies that work with AI-generated content, for example. It is also useful for patrolling users to be able to return to AI-assisted pages when checking. 
:Another thing is that you don't specify which specific templates to use. If you don't specify, everyone will use whatever templates they want and it will lead to chaos. Moreover, who is to determine that the templates are ''not unduly distracting or alarming''? As I wrote above, Wikiversity's policy should be clear. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:56, 19 March 2026 (UTC)

== Publicly available link - risk of link-rot? ==

Is there a risk that the statement:

''&lt;big&gt;The origin of the text must be clearly indicated in the edit summary and ideally include a publicly available link to the chatbot conversation&lt;/big&gt;''

may be problematic in the future if the links go bad (see [[wikipedia:Link_rot|link rot]])? Does the internet archive regularly comb the link of chatbot conversations? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:19, 23 March 2026 (UTC)

:@[[User:IanVG|IanVG]] There may well be link rot over time but linking to the conversation is still better than not linking so that contributions and their sources are reviewable at least until the link does rot -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:48, 24 March 2026 (UTC)

== Mandatory link to chatbot conversation? ==


I'm not sure I'm a fan of (in bold) from the first acceptability requirement that states:&lt;blockquote&gt;The origin of the text must be clearly indicated in the edit summary and '''ideally include''' a publicly available link to the chatbot conversation&lt;/blockquote&gt;Why isn't the requirement strict? Why don't we make the link to the chat mandatory? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:24, 23 March 2026 (UTC)

:@[[User:IanVG|IanVG]] linking to the chat could be made mandatory (and would be better scholarship) but not all LLMs provide a way to publicly link to chats, so such a policy would restrict what AI tools could be used -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 11:45, 24 March 2026 (UTC)
::In true &quot;conversation&quot; cases a pastebin service such as https://paste.toolforge.org/ may be usable. As long as the text is legible as a transcript of the conversation it would be good for scholarship. Would not be usable for cases where AI is used as an &quot;auto-complete&quot; tool like GitHub copilot or Claude Code working on text file containing the wikitext source code though. [[User:Artoria2e5|Artoria2e5]] ([[User talk:Artoria2e5|discuss]] • [[Special:Contributions/Artoria2e5|contribs]]) 05:27, 16 April 2026 (UTC)
:::Agree. I've changed &quot;where available&quot; to &quot;(or a copy of the transcript)&quot;. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:33, 16 April 2026 (UTC)
:Lets keep it simple. Do we really need that link. Patrole is not able to controll all recent changes, who will be patrolling this? I would '''leave it as it is or on request'''. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:37, 25 March 2026 (UTC)
::Patrol can focus on the content itself, not the link. The link is like providing a source code to some media on Commons: good for knowing how it's made and for when modifications are needed. [[User:Artoria2e5|Artoria2e5]] ([[User talk:Artoria2e5|discuss]] • [[Special:Contributions/Artoria2e5|contribs]]) 05:25, 16 April 2026 (UTC)
:::I'd love to have an on-wiki LLM that we can tweak to do what we want here: which is roughly a condensed summary of the prompts and transcript leading up to the final output.  Often the full transcript includes many rounds of iteration and modification, and it will be many times longer than the final output including duplication of that output itself.  (this is the 'default' downloadable transcript where one is available)
:::I think the right request should be &quot;prompt and model&quot; and &quot;link to transcript where possible&quot; to avoid the duplication of the output. &lt;span style=&quot;padding:0 2px 0 2px;background-color:white;color:#bbb;&quot;&gt;&amp;ndash;[[User:Sj|SJ]][[User Talk:Sj|&lt;span style=&quot;color:#ff9900;&quot;&gt;+&lt;/span&gt;]]&lt;/span&gt; 18:34, 1 May 2026 (UTC)

== Superseding the first policy proposal ==

Because the first policy proposal would distract users without due cause;

Because the first policy proposal would burden editors without due cause;

Because the first policy proposal includes elements that are arbitrarily chosen and not derived from stakeholder benefits;

Because great designs are as simple as possible and no simpler;

Because the first policy proposal has failed to attract proponents;

Because the stated objections to the second policy proposal are based on unsound arguments, [[wikipedia:Straw_man|straw men]], speculation, and [[wikipedia:Ad_hominem|ad hominem]] attacks;

I have superseded the first policy proposal text with the second policy proposal text. [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 18:07, 26 March 2026 (UTC)

:Because we are in the process of improving the proposed policy through consensus, I suggest reverting these wholesale changes and working to iteratively improve it. You have strong opinions and some useful ideas; your input is valued. I appreciate [[Wikiversity:Be bold|being bold]], but community consensus is more likely to be achieved through gradual, collaborative iteration. Alternatively, consider forking the proposal and then the community can evolve two versions and then decide on the preferred approach. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:09, 27 March 2026 (UTC)
::Thanks for these comments and your moderating voice. How do I “fork the proposal”? I would like to present alternative policy text with equal visibility to the legacy policy proposal text so that there can be an informed and skillful dialogue leading toward a strong consensus.   [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 11:27, 28 March 2026 (UTC)
:::To &quot;fork&quot;, create a target page e.g., [[Wikiversity:Artificial intelligence 2]] e.g., by:
:::# Manual fork (copy and paste) - but loses edit history
:::# Export/import fork (use [[Special:Export]] and [[Wikiversity:Import]] to copy an original page and retain its edit history) - needs admin rights for import
:::# Or create an alternative policy proposal by starting from scratch
:::See also [[Wikiversity:Productive forking and tailoring is encouraged]] -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 01:23, 29 March 2026 (UTC)
:None of these rationales are based on evidence and/or just completely false (ex, &quot;the first policy proposal has failed to attract proponents&quot; when multiple people have supported the policy as is on the Colloquium). I've removed your edit and I ask you not to do that again. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:03, 27 March 2026 (UTC)
::You are obviously passionate about this issue, and we have differing points of view. Perhaps we can [[Transcending Conflict|transcend conflict]] and find [[Finding Common Ground|common ground]]. I suggest you develop a [[Creating Wikiversity Courses|Wikiversity course]] called something like “Uses and Abuses of Artificial Intelligence.” This will provide all of us with a well-considered basis for developing a policy.  [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 19:20, 27 March 2026 (UTC)
:::Common ground sounds great. I've edited the current (original) draft a little to emphasise adoption of good scholarly practice (e.g., transparency) above specific requirements but also softened the requirement for the AI template to be displayed only for pages with a significant amount of AI-generated material. Hopefully this helps at least somewhat to address some of Lee's concerns. Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 10:54, 28 March 2026 (UTC)

== The Single Mandated Template Needs to Become More Flexible ==

The presently proposed policy mandates the use of a single AI-Generated template for a wide range of AI uses. The scope of the policy identifies a broad range of AI usage, from grammar checkers to generation of extensive text passages. These various uses bear little or no similarity from the user’s perspective. More flexibility, more subtlety, more nuance is needed. I recommend adding parameters to the single mandated template to identify the nature of the AI usage, or providing a family of templates that editors can choose from to more accurately communicate the variety of AI used.  [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 12:54, 1 April 2026 (UTC)

:I think it would be better to have one template with parameters. As I previously mentioned more templates would create more mess from my perspective. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:11, 1 April 2026 (UTC)
:Yes, template parameters could work well. One parameter could allow a text note to explain how gen-AI was used. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:52, 2 April 2026 (UTC)
:This can probably be done technically. But now I realized that there may be a problem with the correct filling if several people edit one page and use different AI tools. There it is more technically feasible, respectively it may be difficult for the user to fill in these parameters and the template itself, or templates, may take up unnecessary space. Therefore, I think that the template can offer these variants, but it would be better if their filling was optional. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:09, 17 April 2026 (UTC)

== Undue Attention and Distraction ==

Attention is our most precious resource, and it must be directed wisely. The presently mandated template places a large banner at the top of each page, as if use of AI is the most important attribute of the learning resource that the user must direct attention to and be concerned with. However, we are acclimating to the use of AI, much as we have acclimated to the use of pocket electronic calculators and the many other innovative technologies that have arisen over the past several decades. The AI notification must become less distracting. I suggest generating a smaller box that appears in the right-had margin like that produced by the ''essay''template. This will better align the attention attracted by the template to the attention it merits.  [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 12:54, 1 April 2026 (UTC)

:Sounds like a good idea. [[User:Artoria2e5|Artoria2e5]] ([[User talk:Artoria2e5|discuss]] • [[Special:Contributions/Artoria2e5|contribs]]) 05:14, 16 April 2026 (UTC)
:I've [https://en.wikiversity.org/w/index.php?title=Template%3AAI-generated&amp;diff=2804949&amp;oldid=2802022 simplified] the {{tl|AI-generated}} message.
:Note that this Wikiversity template is minimalistic compared to sister project equivalents e.g.,
:* [[Template:AI-generated|Template:AI-generated]] (Wikibooks)
:* [[Template:AI-generated|Template:AI-generated]] (Wikipedia)
:-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:23, 16 April 2026 (UTC)
:Yes, this could be, see: [[User:Juandev/T/QA AI contribution]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:06, 17 April 2026 (UTC)

== What problem is being addressed? ==

I recommend we be clear and explicit about the problem, real or perceived, that this policy is intended to address. What are the unmet needs of the users? What are the unmet needs of the editors that need to be addressed by such a policy? We can only rationally evaluate alternative polices in the context of know user and editor needs. Until we understand the users’ needs, and the editors’ needs it is premature to propose a policy.  [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 12:55, 1 April 2026 (UTC)

:'''The basic problem we are solving here is the speed of generating such content.'''
:Qualitatively, texts created with the help of artificial intelligence are equal to texts without the use of AI. There is a range of contributions by quality: high-quality texts, average, and bad ones.
:Methods developed for text control, which were developed on Wikimedia projects, can fail in the case of quickly generated text in that the project will be overwhelmed with such content very quickly that some methods of control will fail and then the quality of the project will decrease.
:That is, we are looking for new solutions to prevent this and one of such solutions is to
:#''remind editors to check the LLM output'',
:#''notify others that the content was created using AI''.
:The control methods used so far are based on creating categories of edits. However, for non-AI contributions categories are recognizable (or can be recognized by a computer program), for AI contributions, I am not aware of a recognition method, so I think it is appropriate for the creator to '''voluntarily report''' AI was used. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:38, 1 April 2026 (UTC)
::Why is increased production speed a problem?
::There is no strong link between production speed and product quality. Electronic calculators and electronic spreadsheet increase both speed and accuracy. Word processors, spell checkers, grammar checkers, on-line dictionaries and thesauruses also increase speed and accuracy. Large Language Models can be used and abused in a wide variety of ways. Certainly, using a LLM to proofread copy, suggest alternative word choice, suggesting rewrites for an awkward sentence and other uses increase the quality of the final product. 
::Although the basic problem is stated as “speed of generation” perhaps the problem to be addressed is the quality of the resulting text.
::The quality of Wikiversity learning resources depends on many factors including curriculum design, topic choice, pedagogical approach, vocabulary choice, prerequisite assumptions, and of course, the factual accuracy of propositional sentences. LLM use pertains to only a fraction of these considerations. Do we have reliable evidence that when LLM’s are used skillfully they are less accurate than material written by the typical Wikiversity editor? 
::Existing Wikiversity policies address the accuracy of the content contributed. As we propose development of AI-specific policies, we need to be clearer and more accurate regarding the problem we are addressing. We need to be more parsimonious in developing policy to address actual problems.  [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 20:08, 9 April 2026 (UTC)
:::@[[User:Lbeaumont|Lbeaumont]] So the problem is still speed. You say, let's solve the problem when it occurs, but here it may happen that when the problem occurs, we will no longer be able to solve it, because we will be flooded with problems and we will not know where they are in that volume. That means, you find one problem in one page and youll figure out its in all pages, but you cannot determine which ones are thos pages. In other words, if the patrol team is now weakened, then it will be totally paralyzed when LLM texts or problems are arose – it will have many times more work than if the text created by LLM was marked and categorized. And secondly, the problem that LLM brings may not be revealed for a very long time. At the same time, it has long been known that LLM hallucinates and, for example, in GPT version 5, the hallucinations have increased slightly compared to version 4o.
:::In other words, I say let's mark and categorize pages with a significant LLM contribution. The marking informs the reader who the author is (correct marking of authorship is the gold standard in Western culture) and let's categorize them so that in the event of a problem we are able to catch the problem. (text created with GT, proofreading human) [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 21:53, 12 April 2026 (UTC)
::::@[[User:Juandev|Juandev]] Help me understand the phrase &quot;patrol team&quot;. Thanks. [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 12:53, 15 April 2026 (UTC)
:::::See [[:w:Wikipedia:Recent changes patrol]] @[[User:Lbeaumont|Lbeaumont]]. Even though we don't have an informative page about this on Wikiversity, it actually naturally exists on all projects. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 11:55, 17 April 2026 (UTC)
::::::Thanks, this provides a valuable service. I was not aware how organized this work is. Do patrol team members coordinate, distributing the work to ensure coverage with minimal duplication? Do they leave some marker (“Kilroy was here”) to inform the original editor and subsequent patrol team members that any particular page was scanned?  [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 13:33, 19 April 2026 (UTC)
:::::::Yes, today there is a feature to check every revision and let it know to others via Patrol button. [https://en.wikiversity.org/wiki/Special:Log?type=patrol&amp;user=&amp;page=&amp;excludetempacct=1&amp;wpdate=&amp;tagfilter=&amp;wpfilters%5B%5D=newusers&amp;wpFormIdentifier=logeventslist See] @[[User:Lbeaumont|Lbeaumont]]. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:05, 5 May 2026 (UTC)
:Agree with @[[User:Juandev|Juandev]] that the basic problem to tackle is to have some sort of control/filter over excessive, low-quality gen-AI content being contributed in a way that would diminish rather than enhance the educational value of this project.
:In the first phase, we've just waited to see what happens. And recently there have been some instructive instances of low-quality gen-AI content so that has helped inform our ideas as have the approaches taken by other sister projects.
:I think it is good scholarly practice to inform readers about the genesis of text. Wiki does this typically very well through edit summaries. So, this should ideally be used to communicate and show specific gen-AI chat sources.
:And a gen-AI info box allows pages with significant gen-AI content to be flagged to readers and categorised. 
:Above all, for me, this is about intellectual honesty. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:48, 2 April 2026 (UTC)

== Tangential: style ==

IMHO the biggest issue with &quot;AI-generated content&quot; is the long-winded, low-information-density writing style it defaults to, complete with unnecessary use of lists over prose and boldening of text. The thing is that it does not take an AI to write like that -- humans who think lazily do many of the same things, especially when influenced by the AI writing-style in everyday conversations. Humans write text full of hot air all the time, complete with references that they did not read. Every fault we have seen in an AI has an analogue in some group of academic humans.

While tagging AI-generated content will aid in the detection of mass-manufactured hot air, it will not address the &quot;artisanal&quot; hot air lovingly typed by some human fingers. The [[WV:MOS]] should be expanded to cover some of these issues. [[User:Artoria2e5|Artoria2e5]] ([[User talk:Artoria2e5|discuss]] • [[Special:Contributions/Artoria2e5|contribs]]) 05:24, 16 April 2026 (UTC)

:Agree. IMHO, a lot of human writing could be improved by running it through a language model.
:Feel free to suggest MOS improvements: [[Wikiversity talk:Manual of Style]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 05:29, 16 April 2026 (UTC)
:Wikiversity is not an encyclopedia, nor a repository of professional texts, although we also store professional texts here. I wonder if blank pages or lengthy narration are not more a methodology within a certain course. In short, Wikiversity cannot be judged through the lens of Wikipedia. Wikipedia, Wiktionary, Wikibooks and other content focus projects can be judged through the lens of Wikipedia, but I find it debatable for Wikiversity @[[User:Artoria2e5|Artoria2e5]].
:Otherwise, for ordinary inflated or less quality text, there are control mechanisms such as [[:w:Wikipedia:Recent changes patrol|patrol]] and monitoring of watchlist, which could also be applied to text created by artificial intelligence. The problem with AI texts, however, is the speed of creation, so I would be in favor of adopting this special policy that would allow such text to be categorized. Ones its categorized, it could be easily checked. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:10, 17 April 2026 (UTC)
::{{quote|IMHO the biggest issue with &quot;AI-generated content&quot; is the long-winded, low-information-density writing style it defaults to, complete with unnecessary use of lists over prose and boldening of text.}}
::I think this could be developed into a useful resource where we suggest ways to type shorter text to a &quot;Quasi-AI&quot;, ie. &quot;Always reply with max 20 word responses to my input and if you type more words then your next response will be more limited each time I notice it&quot; (something like this, so the instructions  to the &quot;Quasi-AI&quot; and to any human reading the input understands what the goal is). That way anybody reading the text doesn't have to read a jungle of text. I myself have a problem with typing a lot of text and I believe &quot;Quasi-AI&quot; could be human's next best friend.(my reference is regarding &quot;The dog is human's best friend&quot;). This text was 100% human generated.
:[[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:29, 2 May 2026 (UTC)

== I'm worried about my text becoming too much ==

If my input / output archive of AI Prompts becomes too much or if you have an idea for how I can better save space, please let me know on my talk page. I'm trying to use the &quot;quasi-AI&quot; tools both to motivate me and to become bolder(&quot;be bold&quot;) in my editing.

I already know text takes up very little space but still I'm worried for some reason...maybe worried to try new things, like documenting extensively what I type to the &quot;Auasi-AI&quot; and I usually copy and text a lot of repeating texts so I'm worried about the text size building up &quot;exponentially&quot;.

Part of the reason for posting this is because sometimes or most of the time when I'm in a specific &quot;psychological mood&quot; I'm feeling like all the ground is just thin ice. So maybe today I want to contribute but I feel like I'm on thin ice. All might just be an illusion that I have to go through. If anyone got any advice for any resource you got(maybe even more preferable if it is &quot;AI-generated&quot;) please let me know if it can help contributors/learners become bolder. 

I hope that in the future we will accept &quot;Quasi-AI&quot; edits so much more so that we may even ask people who may struggle with understanding and outputting human text might be encouraged to use a &quot;Quasi-AI&quot; for their output, ie. my father asked me to use a &quot;Quasi-AI&quot; because he can't stand my &quot;blathering&quot; because he prefers &quot;spartanic minimalism&quot; and saying more with less text. This text was 100% human generated. [[User:ThinkingScience|ThinkingScience]] ([[User talk:ThinkingScience|discuss]] • [[Special:Contributions/ThinkingScience|contribs]]) 10:11, 2 May 2026 (UTC)

:BTW, what is quasi-AI text @[[User:ThinkingScience|ThinkingScience]]? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 12:11, 5 May 2026 (UTC)

== Burden of Proof ==

The (only) provided template for indicating AI use makes the specific claim “This resource includes substantial content generated by artificial intelligence”. Use of this template should be limited to instances where the editor introducing the template has reliable evidence that the AI use is indeed substantial, and not simply incidental. The editor introducing that template bears the burden of proof that AI use is indeed substantial. In the case of dispute between the originating editor, who made use of AI, and any reviewing editor, who is alleging substantial use of AI, the benefit of the doubt goes to the originating editor, rather than to the reviewing editor.  [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 14:46, 6 May 2026 (UTC)</text>
      <sha1>embtk7n26up267jq5rwxa6t924160uv</sha1>
    </revision>
  </page>
  <page>
    <title>Just sustainability transitions: a living review</title>
    <ns>0</ns>
    <id>326060</id>
    <revision>
      <id>2807807</id>
      <parentid>2807799</parentid>
      <timestamp>2026-05-06T13:31:28Z</timestamp>
      <contributor>
        <username>Amélie E. Pereira</username>
        <id>3042711</id>
      </contributor>
      <comment>/* Study types */</comment>
      <origin>2807807</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="43347" sha1="9ha50kr578o3049oqfqx0ziig0tyz6q" xml:space="preserve">== Introduction ==

=== Definition of living review ===
The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked&lt;ref name=&quot;Why1&quot;&gt;{{Cite Q |Q40040379 }}&lt;/ref&gt;. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot;&gt;{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}&lt;/ref&gt;&lt;ref name=&quot;:6&quot;&gt;{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}&lt;/ref&gt;. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication&lt;ref name=&quot;Why1&quot;/&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;. Living systematic reviews exists to address this common problem&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/&lt;/ref&gt;. It is therefore particularly useful in rapidly evolving fields of research&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;, such as just transition. 

[[wikidata:Q33002955|Knowledge graphs]], a structured representation of knowledge in the form of a graph, linked together by relationships that encode explicit meanings between these entities, are very suitable for conducting living systematic reviews&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;Fotopoulou&quot;&gt;{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.&lt;/ref&gt;. Advances in AI could render certain older methodological types of living systematic reviews obsoletes&lt;ref&gt;{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}&lt;/ref&gt;, as IA are useful to extract, filter and classify datas&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;. [[Large language models]] (LLM) are &quot;on the rise&quot; (2025), but &quot;not yet ready for use&quot;&lt;ref&gt;{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}&lt;/ref&gt;.

The living review method relevant for just transition because it includes topic such as energy democracy which necessitate transdisciplinarity and consolidation of fragmented literature&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;.

===  Definitions of just transition : ===
* «a fair and equitable process of moving towards a post-carbon society’. »&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}&lt;/ref&gt;.


The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions&lt;ref name=&quot;:0&quot; /&gt;. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions&lt;ref name=&quot;:0&quot; /&gt;. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research &amp; Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}&lt;/ref&gt; and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 
 
=== Definition of Procedural justice ===
Procedural justice is about the fairness of decision-making processes related to transitions&lt;ref name=&quot;:4&quot; /&gt; such as the inclusion of those impacted by these decisions&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}&lt;/ref&gt;. Procedural justice can include issues of community and citizen participation in decision making, their political representation their consultation or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities&lt;ref&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. For example, the participation of affected communities in decisions related to the construction of new infrastructures&lt;ref name=&quot;:0&quot; /&gt;.     

== Methodology ==

=== Wikidata and the semantic web ===&lt;!-- Add introduction to what wikidata is and how the triplet works in a pedagogical manner
 --&gt;

=== Database search ===
We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. 
 
{| class=&quot;wikitable&quot;
|+
!Keywords search
!Database
!Search date
!Filters
!Number of results
|-
|(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) 
|Web of Science (all databases, all dates)
|December 2025
|Document type: Review Article
|362
|}

=== Article screening ===
Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were 

* Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...)
* Not literature reviews (e.g. review of policies, initiatives, cases, review notes, book review...)
* Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors or included justice consideration only in “future research” suggestions
* Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy
* Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper

=== Importing selected articles into Wikidata ===
To import the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in the database. Next we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the &quot;export to Wikidata QuickStatements&quot; function of Zotero and use the QuickStatements tool to add them to Wikidata.

Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well  because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|&quot;Merge&quot; gadget]] on Wikidata. We checked manually for duplicated statments in those items.

=== Article classification through meta-data enrichement ===&lt;!-- Add : What is meta-data enrichement --&gt;

Existing review try to classify existing articles according to various criteria such as industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt).&lt;ref name=&quot;:5&quot; /&gt; We selected the most relevant properties in Wikidata to reflect these classifications : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe its geographical focus.

==== Main subjects ====
We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : 
{| class=&quot;wikitable&quot;
|+
!Qid
!Main topic
!Description
|-
|[[d:Q42377797|Q42377797]]
|acceptability
|characteristic of a thing being subject to acceptance for some purpose
|-
|[[d:Q2798912|Q2798912]]
|accountability
|concept of responsibility in ethics, governance and decision-making
|-
|[[d:Q421953|Q421953]]
|actor–network theory
|theory within social science
|-
|[[d:Q84459973|Q84459973]]
|affordability
|
|-
|[[d:Q185836|Q185836]]
|age of a person
|time elapsed since a person was born
|-
|[[d:Q4764988|Q4764988]]
|animal studies
|field in which animals are studied in a variety of cross-disciplinary ways
|-
|[[d:Q4338318|Q4338318]]
|awareness
|state or ability to perceive, to feel, or to be conscious of events, objects, or sensory patterns
|-
|[[d:Q4930066|Q4930066]]
|blue carbon
|carbon captured by the world's coastal ocean ecosystems
|-
|[[d:Q430460|Q430460]]
|capability approach
|economic theory
|-
|[[d:Q7569|Q7569]]
|child
|human between birth and puberty
|-
|[[d:Q4116870|Q4116870]]
|civic engagement
|individual or group activity addressing issues of public concern
|-
|[[d:Q125928|Q125928]]
|climate change
|human-caused changes to climate on Earth
|-
|[[d:Q260607|Q260607]]
|climate change adaptation
|process of adjustment to actual or expected climate change and its effects, seeking to moderate or avoid harm or exploit beneficial opportunities
|-
|[[d:Q1291678|Q1291678]]
|climate justice
|term linking the climate crisis with environmental and social justice
|-
|[[d:Q2270945|Q2270945]]
|co-creation
|product or service design process in which input from consumers plays a central role
|-
|[[d:Q16972712|Q16972712]]
|co-design
|approach to design attempting to actively involve all stakeholders
|-
|[[d:Q16324410|Q16324410]]
|coproduction
|product or service design process in which input from consumers plays a central role
|-
|[[d:Q11024|Q11024]]
|communication
|act of conveying intended meaning
|-
|[[d:Q177634|Q177634]]
|community
|social unit of human organisms who share common values
|-
|[[d:Q5154673|Q5154673]]
|community choice aggregation
|alternative energy supply system
|-
|[[d:Q113514984|Q113514984]]
|community energy
|delivery of community-led renewable energy, energy demand reduction and energy supply projects
|-
|[[d:Q65807646|Q65807646]]
|community participation
|The taking part by members of a community in decisionmaking processes related to the development of their community
|-
|[[d:Q188843|Q188843]]
|cosmopolitanism
|ideology that all human beings belong to a single community, based on a shared morality
|-
|[[d:Q11693783|Q11693783]]
|decarbonization
|change of economy, especially of energy industries, towards lower carbon dioxide emissions
|-
|[[d:Q284289|Q284289]]
|deliberative democracy
|form of democracy focusing on consensus
|-
|[[d:Q7174|Q7174]]
|democracy
|form of government
|-
|[[d:Q552284|Q552284]]
|distributive justice
|concept of the socially just allocation of goods
|-
|[[d:Q1230584|Q1230584]]
|diversity
|concept in sociology and political studies
|-
|[[d:Q1049066|Q1049066]]
|ecological economics
|research field on the interdependence of human economies and natural ecosystems
|-
|[[d:Q8134|Q8134]]
|economics
|social science that studies the production, distribution, and consumption of goods and services
|-
|[[d:Q868575|Q868575]]
|empowerment
|providing increased autonomy
|-
|[[d:Q295865|Q295865]]
|ecosystem service
|benefits created by nature, forests and environmental systems
|-
|[[d:Q138359220|Q138359220]]
|energy citizenship
|involvement of citizens in energy-related decisions
|-
|[https://www.wikidata.org/w/index.php?title=Q131444737&amp;redirect=no Q131444737]
|community energy
|[redirection]
|-
|[[d:Q16869822|Q16869822]]
|energy consumption
|amount of energy or power used
|-
|[[d:Q1358789|Q1358789]]
|senior
|elderly person
|-
|[[d:Q14944319|Q14944319]]
|energy democracy
|concept in environmental justice movement
|-
|[[d:Q192704|Q192704]]
|energy efficiency
|ratio between the useful energy output and the input of a machine
|-
|Q24965464
|
|
|-
|Q1805337
|
|
|-
|Q1341244
|
|
|-
|Q3406659
|
|
|-
|Q117091181
|
|
|-
|Q3456219
|
|
|-
|Q2700433
|
|
|-
|Q837718
|
|
|-
|Q795757
|
|
|-
|Q795757
|
|
|-
|Q1479527
|
|
|-
|Q771773
|
|
|-
|Q56395513
|
|
|-
|Q5465532
|
|
|-
|Q4421
|
|
|-
|Q48277
|
|
|-
|Q1553864
|
|
|-
|Q8458
|
|
|-
|Q11376059
|
|
|-
|Q103817
|
|
|-
|Q113561794
|
|
|-
|Q770480
|
|
|-
|Q17142211
|
|
|-
|Q1516555
|
|
|-
|Q6316391
|
|
|-
|Q366139
|
|
|-
|Q3027857
|
|
|-
|Q59679511
|
|
|-
|Q43619
|
|
|-
|Q127514833
|
|
|-
|Q13023682
|
|
|-
|Q728646
|
|
|-
|Q3907287
|
|
|-
|Q9357091
|
|
|-
|Q265425
|
|
|-
|Q25107
|
|
|-
|Q442100
|
|
|-
|Q7249406
|
|
|-
|Q7257735
|
|
|-
|Q541936
|
|
|-
|Q6142016
|
|
|-
|Q10509953
|
|
|-
|Q12705
|
|
|-
|Q56510941
|
|
|-
|Q1165392
|
|
|-
|Q4414036
|
|
|-
|Q17152351
|
|
|-
|Q187588
|
|
|-
|Q264892
|
|
|-
|Q34749
|
|
|-
|Q2930198
|
|
|-
|Q125359881
|
|
|-
|Q219416
|
|
|-
|Q131201
|
|
|-
|Q7649586
|
|
|-
|Q69883
|
|
|-
|Q920600
|
|
|-
|Q3376054
|
|
|-
|Q107389921
|
|
|-
|Q7981051
|
|
|-
|Q467
|
|
|-
|Q188867
|
|
|-
|Q1038171
|
|
|}
&lt;!-- include all below items using the wikidata link template
 --&gt;

Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords.

==== Study types ====
Our review included only litterature reviews. We first read abstracts to identify all the [https://angryloki.github.io/wikidata-graph-builder/?item=Q2412849&amp;property=P279&amp;mode=reverse different types of litterature reviews] present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved these method items using the methodological references cited in the reviewed papers. The types of reviews were : 
{| class=&quot;wikitable&quot;
|+
!Qid
!Study type
!Description
|-
|[[d:Q603441|Q603441]]
|bibliometrics
|statistical analysis of written publications, such as books or articles
|-
|[[d:Q472342|Q472342]]
|scientometrics
|study of measuring and analysing science, technology and innovation
|-
|[[d:Q815382|Q815382]]
|meta-analysis
|statistical method that summarizes data from multiple sources
|-
|[[d:Q1504425|Q1504425]]
|systematic review
|publication type, study that gathers, analyzes, and communicates the results of research and information on a topic
|-
|[[d:Q2412849|Q2412849]]
|literature review
|process of information search and text of a review article (Q7318358), which includes the current knowledge including substantive findings, as well as theoretical and methodological contributions to a particular topic
|-
|[[d:Q6822263|Q6822263]]
|meta-regression
|statistical tool used in meta-analyses
|-
|[[d:Q7301211|Q7301211]]
|realist evaluation
|[...]
|-
|[[d:Q17007303|Q17007303]]
|combinatorial meta-analysis 
|[...]
|-
|[[d:Q70470634|Q70470634]]
|network meta-analysis
|meta-analysis of randomized trials in which estimates of comparative treatment effects are visualized and interpreted from a network of interventions
|-
|[[d:Q101116078|Q101116078]]
|scoping review
|search for concepts by mapping the language and data which surrounds those concepts and adjusting the search method iteratively to synthesize evidence and assess the scope of an area of inquiry
|-
|[[d:Q110665014|Q110665014]]
|narrative review
|type of literature review, without structured method of retrieval and analysis
|-
|[[d:Q137174203|Q137174203]]
|conceptual review
|academic research aiming to review existing concepts and definitions in the litterature
|-
|[[d:Q137174450|Q137174450]]
|critical review
|type of literature review analysing strenghts, major contributions, mistakes and neglected issues in an academic field of research
|-
|[[d:Q137209848|Q137209848]]
|integrative literature review
|type of literature review
|-
|[[d:Q110665014|Q137211242]]
|narrative review
|type of literature review, without structured method of retrieval and analysis
|}&lt;!-- include all below items using the wikidata link template
 --&gt;
[Include list and description of types of litterature reviews]

Then, we added the {{Wikidata entity link|P8363}} of each articles based on the abstract and method sections. In case of doubt, we compared our interpretation.

==== Research site ====
When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article &quot;{{Wikidata entity link|Q137901202}}&quot; focused on {{Wikidata entity link|Q132959}}.

==== Results ====
[insert table about the sample]

=== Knowledge modelling ===
Concept maps can be a powerful literature review tool&lt;ref&gt;{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}&lt;/ref&gt; allowing to synthetize theoretical statements about relationship between concepts&lt;ref&gt;{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}&lt;/ref&gt;. In the present study, we explored how concept map can be used to model the knowledge present in the paper we selected. 

[define knowledge modelling]

==== Conceptual modelling ====
We first reflected on what kind of wikidata properties could be used to represent concepts and theories in wikidata. Capturing the content of a concept is not straightforward and there are various approaches coming from psychology and philosophy on the matter&lt;ref&gt;{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}&lt;/ref&gt; we summarize these approaches below and examine which wikidata properties exist to represent them.
* Definition: the content of a concept can be formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}...
* Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}}.
* Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}.
* Essence: the content of a concept is &quot;something&quot; deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can create it before really knowing what all its properties will be.
* Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}.

==== Thematic networks ====
[[File:Thematic network example.jpg|thumb|547x547px|Structure of a thematic network (Source: Attride-Stirling 2001)]]
A thematic network is “simply a way of organizing a thematic analysis of qualitative data”&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}&lt;/ref&gt;. It is compatible with classical coding strategies such as [[grounded theory]]&lt;ref&gt;{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}&lt;/ref&gt;. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data&lt;ref name=&quot;:7&quot; /&gt;. The principle is to assemble basic themes into more general themes.   

Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding (e.g. grounded theory) to identify themes and sub-themes.   

However, the nature of the relationship between these various themes and sub-themes is often not specified.  

*

==== Causal networks ====
The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers sometime present models with boxes and arrows representing correlations and/or causations&lt;ref&gt;{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}&lt;/ref&gt;. In qualitative research, building grounded theory models is about &quot;[accounting] for not only all the major emergent concepts, themes, and dimensions, but also for their dynamic interrelationships. Speaking in classic boxes-and-arrows terms, this process amounts to assembling the constellation of boxes with a special focus on the arrows.&quot;&lt;ref&gt;{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}&lt;/ref&gt; Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can &quot;feedback&quot; (two variables can influence each other)&lt;ref&gt;{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}&lt;/ref&gt;.

Wikidata includes several properties to describe causal relationships:
* {{Wikidata entity link|P828}}
* {{Wikidata entity link|P1542}}
* {{Wikidata entity link|P1537}}
* {{Wikidata entity link|P1479}} : it is difficult to identify single causes for social phenomenons, many factors having an effect on the subject item will likely be contributing factors

==== Modelling concepts ====
To model concepts related to just transition. We read the selected papers and used them as source to build a knowledge graph in wikidata. For example, the paper {{Wikidata entity link|Q137901182}} mention &quot;Energy democracy is both an ideal and a process&quot;&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;, we thus entered the wikidata statement {{Wikidata entity link|Q14944319}} is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q840396}}, using the paper as source.

Challenges :
*{{Wikidata entity link|P31}}: concepts have a dual nature because they designate at the same time an idea and the entity that this idea represent.
* {{Wikidata entity link|P3712}}: concepts do not have goals in themselves, but the reality they represent can have goals
* wikidata current items are not really suited to model &quot;meta-research&quot; statements. For example, modelling the idea tha the literature on energy democracy is fragmented would require creating an item representing the energy democracy literature, not just energy democracy in general. Similarly, it can be difficult to model the chronological evolution of the definition of an idea (although it could be technically possible).
* To distinguish the causes of the concepts/discourses and the causes of the phenomenon itself, we used {{Wikidata entity link|P3938}} to indicate the origins of the concept or the movements promoting it.
* To distinguish goals from the process to reach it, we used {{Wikidata entity link|P3712}} to describe ideals and {{Wikidata entity link|P2670}} to describe processes.
* Wikidata does not seem to be the best tool to model quantitative statements, for example, the paper {{Wikidata entity link|Q137901196}} states that &quot;9.8% of the final energy consumed in developing countries comes from modern renewable energy sources&quot;. Including energy data in Wikidata require using or creating specific properties (e.g. {{Wikidata entity link|P6826}})
*

=== Writing ===
To cite articles we used the [[Template:Cite Q|Cite Q template.]] Each reference is an item in Wikidata and the template retrieve the necessary data to generate the citation references below.

== Data == 
{| class=&quot;wikitable sortable&quot;
! QID !! Year !! DOI !! Title
|-
| [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review
|-
| [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review
|-
| [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review
|-
| [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter?
|-
| [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset.
|-
| [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies?
|-
| [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection
|-
| [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development
|-
| [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research
|-
| [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition
|-
| [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning
|-
| [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review
|-
| [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view
|-
| [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory
|-
| [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries
|-
| [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review
|-
| [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions
|-
| [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies
|-
| [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes
|-
| [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation
|-
| [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives
|-
| [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies
|-
| [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda
|-
| [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice?
|-
| [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review
|-
| [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research
|-
| [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape
|-
| [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models
|-
| [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review
|-
| [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions
|-
| [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions
|-
| [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation
|-
| [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings
|-
| [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda
|-
| [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review
|-
| [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework
|-
| [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende
|-
| [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa
|-
| [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities
|-
| [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion
|-
| [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review
|-
| [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights
|-
| [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review
|-
| [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations
|-
| [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance
|-
| [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions
|-
| [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review
|-
| [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice
|-
| [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice
|-
| [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review
|-
| [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review
|-
| [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions
|-
| [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition
|-
| [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy
|-
| [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends
|-
| [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience
|}


== References ==
{{References}}</text>
      <sha1>9ha50kr578o3049oqfqx0ziig0tyz6q</sha1>
    </revision>
    <revision>
      <id>2807816</id>
      <parentid>2807807</parentid>
      <timestamp>2026-05-06T14:22:30Z</timestamp>
      <contributor>
        <username>Amélie E. Pereira</username>
        <id>3042711</id>
      </contributor>
      <comment>/* Main subjects */</comment>
      <origin>2807816</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="44480" sha1="j9jb9thhhjgkmay995ruthikjm5spfo" xml:space="preserve">== Introduction ==

=== Definition of living review ===
The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked&lt;ref name=&quot;Why1&quot;&gt;{{Cite Q |Q40040379 }}&lt;/ref&gt;. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot;&gt;{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}&lt;/ref&gt;&lt;ref name=&quot;:6&quot;&gt;{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}&lt;/ref&gt;. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication&lt;ref name=&quot;Why1&quot;/&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;. Living systematic reviews exists to address this common problem&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;Why4&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;&lt;ref name=&quot;:2&quot;&gt;https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/&lt;/ref&gt;. It is therefore particularly useful in rapidly evolving fields of research&lt;ref name=&quot;Why1&quot; /&gt;&lt;ref name=&quot;:6&quot; /&gt;, such as just transition. 

[[wikidata:Q33002955|Knowledge graphs]], a structured representation of knowledge in the form of a graph, linked together by relationships that encode explicit meanings between these entities, are very suitable for conducting living systematic reviews&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;Fotopoulou&quot;&gt;{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.&lt;/ref&gt;. Advances in AI could render certain older methodological types of living systematic reviews obsoletes&lt;ref&gt;{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}&lt;/ref&gt;, as IA are useful to extract, filter and classify datas&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}&lt;/ref&gt;. [[Large language models]] (LLM) are &quot;on the rise&quot; (2025), but &quot;not yet ready for use&quot;&lt;ref&gt;{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}&lt;/ref&gt;.

The living review method relevant for just transition because it includes topic such as energy democracy which necessitate transdisciplinarity and consolidation of fragmented literature&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;.

===  Definitions of just transition : ===
* «a fair and equitable process of moving towards a post-carbon society’. »&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}&lt;/ref&gt;.


The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions&lt;ref name=&quot;:0&quot; /&gt;. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions&lt;ref name=&quot;:0&quot; /&gt;. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice&lt;ref name=&quot;:3&quot;&gt;{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research &amp; Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}&lt;/ref&gt;&lt;ref name=&quot;:1&quot;&gt;{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}&lt;/ref&gt; and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice&lt;ref name=&quot;:0&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt;&lt;ref name=&quot;:1&quot; /&gt;&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. 
 
=== Definition of Procedural justice ===
Procedural justice is about the fairness of decision-making processes related to transitions&lt;ref name=&quot;:4&quot; /&gt; such as the inclusion of those impacted by these decisions&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}&lt;/ref&gt;. Procedural justice can include issues of community and citizen participation in decision making, their political representation their consultation or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities&lt;ref&gt;{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research &amp; Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}&lt;/ref&gt;. For example, the participation of affected communities in decisions related to the construction of new infrastructures&lt;ref name=&quot;:0&quot; /&gt;.     

== Methodology ==

=== Wikidata and the semantic web ===&lt;!-- Add introduction to what wikidata is and how the triplet works in a pedagogical manner
 --&gt;

=== Database search ===
We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. 
 
{| class=&quot;wikitable&quot;
|+
!Keywords search
!Database
!Search date
!Filters
!Number of results
|-
|(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) 
|Web of Science (all databases, all dates)
|December 2025
|Document type: Review Article
|362
|}

=== Article screening ===
Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were 

* Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...)
* Not literature reviews (e.g. review of policies, initiatives, cases, review notes, book review...)
* Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors or included justice consideration only in “future research” suggestions
* Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy
* Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper

=== Importing selected articles into Wikidata ===
To import the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in the database. Next we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the &quot;export to Wikidata QuickStatements&quot; function of Zotero and use the QuickStatements tool to add them to Wikidata.

Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well  because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|&quot;Merge&quot; gadget]] on Wikidata. We checked manually for duplicated statments in those items.

=== Article classification through meta-data enrichement ===&lt;!-- Add : What is meta-data enrichement --&gt;

Existing review try to classify existing articles according to various criteria such as industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt).&lt;ref name=&quot;:5&quot; /&gt; We selected the most relevant properties in Wikidata to reflect these classifications : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe its geographical focus.

==== Main subjects ====
We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : 
{| class=&quot;wikitable&quot;
|+
!Qid
!Main topic
!Description
|-
|[[d:Q42377797|Q42377797]]
|acceptability
|characteristic of a thing being subject to acceptance for some purpose
|-
|[[d:Q2798912|Q2798912]]
|accountability
|concept of responsibility in ethics, governance and decision-making
|-
|[[d:Q421953|Q421953]]
|actor–network theory
|theory within social science
|-
|[[d:Q84459973|Q84459973]]
|affordability
|
|-
|[[d:Q185836|Q185836]]
|age of a person
|time elapsed since a person was born
|-
|[[d:Q4764988|Q4764988]]
|animal studies
|field in which animals are studied in a variety of cross-disciplinary ways
|-
|[[d:Q4338318|Q4338318]]
|awareness
|state or ability to perceive, to feel, or to be conscious of events, objects, or sensory patterns
|-
|[[d:Q4930066|Q4930066]]
|blue carbon
|carbon captured by the world's coastal ocean ecosystems
|-
|[[d:Q430460|Q430460]]
|capability approach
|economic theory
|-
|[[d:Q7569|Q7569]]
|child
|human between birth and puberty
|-
|[[d:Q4116870|Q4116870]]
|civic engagement
|individual or group activity addressing issues of public concern
|-
|[[d:Q125928|Q125928]]
|climate change
|human-caused changes to climate on Earth
|-
|[[d:Q260607|Q260607]]
|climate change adaptation
|process of adjustment to actual or expected climate change and its effects, seeking to moderate or avoid harm or exploit beneficial opportunities
|-
|[[d:Q1291678|Q1291678]]
|climate justice
|term linking the climate crisis with environmental and social justice
|-
|[[d:Q2270945|Q2270945]]
|co-creation
|product or service design process in which input from consumers plays a central role
|-
|[[d:Q16972712|Q16972712]]
|co-design
|approach to design attempting to actively involve all stakeholders
|-
|[[d:Q16324410|Q16324410]]
|coproduction
|product or service design process in which input from consumers plays a central role
|-
|[[d:Q11024|Q11024]]
|communication
|act of conveying intended meaning
|-
|[[d:Q177634|Q177634]]
|community
|social unit of human organisms who share common values
|-
|[[d:Q5154673|Q5154673]]
|community choice aggregation
|alternative energy supply system
|-
|[[d:Q113514984|Q113514984]]
|community energy
|delivery of community-led renewable energy, energy demand reduction and energy supply projects
|-
|[[d:Q65807646|Q65807646]]
|community participation
|The taking part by members of a community in decisionmaking processes related to the development of their community
|-
|[[d:Q188843|Q188843]]
|cosmopolitanism
|ideology that all human beings belong to a single community, based on a shared morality
|-
|[[d:Q11693783|Q11693783]]
|decarbonization
|change of economy, especially of energy industries, towards lower carbon dioxide emissions
|-
|[[d:Q284289|Q284289]]
|deliberative democracy
|form of democracy focusing on consensus
|-
|[[d:Q7174|Q7174]]
|democracy
|form of government
|-
|[[d:Q552284|Q552284]]
|distributive justice
|concept of the socially just allocation of goods
|-
|[[d:Q1230584|Q1230584]]
|diversity
|concept in sociology and political studies
|-
|[[d:Q1049066|Q1049066]]
|ecological economics
|research field on the interdependence of human economies and natural ecosystems
|-
|[[d:Q8134|Q8134]]
|economics
|social science that studies the production, distribution, and consumption of goods and services
|-
|[[d:Q868575|Q868575]]
|empowerment
|providing increased autonomy
|-
|[[d:Q295865|Q295865]]
|ecosystem service
|benefits created by nature, forests and environmental systems
|-
|[[d:Q138359220|Q138359220]]
|energy citizenship
|involvement of citizens in energy-related decisions
|-
|[https://www.wikidata.org/w/index.php?title=Q131444737&amp;redirect=no Q131444737]
|community energy
|[redirection]
|-
|[[d:Q16869822|Q16869822]]
|energy consumption
|amount of energy or power used
|-
|[[d:Q1358789|Q1358789]]
|senior
|elderly person
|-
|[[d:Q14944319|Q14944319]]
|energy democracy
|concept in environmental justice movement
|-
|[[d:Q192704|Q192704]]
|energy efficiency
|ratio between the useful energy output and the input of a machine
|-
|[[d:Q24965464|Q24965464]]
|energy modeling
|process of building computer models of energy systems in order to analyze them
|-
|[[d:Q1805337|Q1805337]]
|energy policy
|policy addressing energy issues
|-
|[[d:Q1341244|Q1341244]]
|energy poverty
|lack of access to modern energy services
|-
|[[d:Q3406659|Q3406659]]
|energy production
|conversion of energy from a primary source into a form useful to humans
|-
|[[d:Q117091181|Q117091181]]
|energy justice
|subconcept of economic equality
|-
|[[d:Q3456219|Q3456219]]
|energy renovation
|building works aimed at reducing energy consumption and decarbonising the energy sources used
|-
|[[d:Q2700433|Q2700433]]
|energy security
|national security considerations of energy availability
|-
|[[d:Q837718|Q837718]]
|energy storage
|capture of energy produced at one time for use at a later time
|-
|[[d:Q795757|Q795757]]
|energy transition
|long-term structural change towards sustainable energy systems
|-
|[[d:Q1479527|Q1479527]]
|environmental justice
|system of fairness
|-
|[[d:Q771773|Q771773]]
|fairness
|concept in sociology and generally the interaction of society
|-
|[[d:Q56395513|Q56395513]]
|farming system
|method of agricultural production defined by its physical practices and economic characteristics
|-
|[[d:Q5465532|Q5465532]]
|food system
|all processes and infrastructure involved in feeding a population
|-
|Q4421
|
|
|-
|Q48277
|
|
|-
|Q1553864
|
|
|-
|Q8458
|
|
|-
|Q11376059
|
|
|-
|Q103817
|
|
|-
|Q113561794
|
|
|-
|Q770480
|
|
|-
|Q17142211
|
|
|-
|Q1516555
|
|
|-
|Q6316391
|
|
|-
|Q366139
|
|
|-
|Q3027857
|
|
|-
|Q59679511
|
|
|-
|Q43619
|
|
|-
|Q127514833
|
|
|-
|Q13023682
|
|
|-
|Q728646
|
|
|-
|Q3907287
|
|
|-
|Q9357091
|
|
|-
|Q265425
|
|
|-
|Q25107
|
|
|-
|Q442100
|
|
|-
|Q7249406
|
|
|-
|Q7257735
|
|
|-
|Q541936
|
|
|-
|Q6142016
|
|
|-
|Q10509953
|
|
|-
|Q12705
|
|
|-
|Q56510941
|
|
|-
|Q1165392
|
|
|-
|Q4414036
|
|
|-
|Q17152351
|
|
|-
|Q187588
|
|
|-
|Q264892
|
|
|-
|Q34749
|
|
|-
|Q2930198
|
|
|-
|Q125359881
|
|
|-
|Q219416
|
|
|-
|Q131201
|
|
|-
|Q7649586
|
|
|-
|Q69883
|
|
|-
|Q920600
|
|
|-
|Q3376054
|
|
|-
|Q107389921
|
|
|-
|Q7981051
|
|
|-
|Q467
|
|
|-
|Q188867
|
|
|-
|Q1038171
|
|
|}
&lt;!-- include all below items using the wikidata link template
 --&gt;

Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords.

==== Study types ====
Our review included only litterature reviews. We first read abstracts to identify all the [https://angryloki.github.io/wikidata-graph-builder/?item=Q2412849&amp;property=P279&amp;mode=reverse different types of litterature reviews] present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved these method items using the methodological references cited in the reviewed papers. The types of reviews were : 
{| class=&quot;wikitable&quot;
|+
!Qid
!Study type
!Description
|-
|[[d:Q603441|Q603441]]
|bibliometrics
|statistical analysis of written publications, such as books or articles
|-
|[[d:Q472342|Q472342]]
|scientometrics
|study of measuring and analysing science, technology and innovation
|-
|[[d:Q815382|Q815382]]
|meta-analysis
|statistical method that summarizes data from multiple sources
|-
|[[d:Q1504425|Q1504425]]
|systematic review
|publication type, study that gathers, analyzes, and communicates the results of research and information on a topic
|-
|[[d:Q2412849|Q2412849]]
|literature review
|process of information search and text of a review article (Q7318358), which includes the current knowledge including substantive findings, as well as theoretical and methodological contributions to a particular topic
|-
|[[d:Q6822263|Q6822263]]
|meta-regression
|statistical tool used in meta-analyses
|-
|[[d:Q7301211|Q7301211]]
|realist evaluation
|[...]
|-
|[[d:Q17007303|Q17007303]]
|combinatorial meta-analysis 
|[...]
|-
|[[d:Q70470634|Q70470634]]
|network meta-analysis
|meta-analysis of randomized trials in which estimates of comparative treatment effects are visualized and interpreted from a network of interventions
|-
|[[d:Q101116078|Q101116078]]
|scoping review
|search for concepts by mapping the language and data which surrounds those concepts and adjusting the search method iteratively to synthesize evidence and assess the scope of an area of inquiry
|-
|[[d:Q110665014|Q110665014]]
|narrative review
|type of literature review, without structured method of retrieval and analysis
|-
|[[d:Q137174203|Q137174203]]
|conceptual review
|academic research aiming to review existing concepts and definitions in the litterature
|-
|[[d:Q137174450|Q137174450]]
|critical review
|type of literature review analysing strenghts, major contributions, mistakes and neglected issues in an academic field of research
|-
|[[d:Q137209848|Q137209848]]
|integrative literature review
|type of literature review
|-
|[[d:Q110665014|Q137211242]]
|narrative review
|type of literature review, without structured method of retrieval and analysis
|}&lt;!-- include all below items using the wikidata link template
 --&gt;
[Include list and description of types of litterature reviews]

Then, we added the {{Wikidata entity link|P8363}} of each articles based on the abstract and method sections. In case of doubt, we compared our interpretation.

==== Research site ====
When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article &quot;{{Wikidata entity link|Q137901202}}&quot; focused on {{Wikidata entity link|Q132959}}.

==== Results ====
[insert table about the sample]

=== Knowledge modelling ===
Concept maps can be a powerful literature review tool&lt;ref&gt;{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}&lt;/ref&gt; allowing to synthetize theoretical statements about relationship between concepts&lt;ref&gt;{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}&lt;/ref&gt;. In the present study, we explored how concept map can be used to model the knowledge present in the paper we selected. 

[define knowledge modelling]

==== Conceptual modelling ====
We first reflected on what kind of wikidata properties could be used to represent concepts and theories in wikidata. Capturing the content of a concept is not straightforward and there are various approaches coming from psychology and philosophy on the matter&lt;ref&gt;{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}&lt;/ref&gt; we summarize these approaches below and examine which wikidata properties exist to represent them.
* Definition: the content of a concept can be formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}...
* Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}}.
* Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}.
* Essence: the content of a concept is &quot;something&quot; deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can create it before really knowing what all its properties will be.
* Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}.

==== Thematic networks ====
[[File:Thematic network example.jpg|thumb|547x547px|Structure of a thematic network (Source: Attride-Stirling 2001)]]
A thematic network is “simply a way of organizing a thematic analysis of qualitative data”&lt;ref name=&quot;:7&quot;&gt;{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}&lt;/ref&gt;. It is compatible with classical coding strategies such as [[grounded theory]]&lt;ref&gt;{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}&lt;/ref&gt;. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data&lt;ref name=&quot;:7&quot; /&gt;. The principle is to assemble basic themes into more general themes.   

Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding (e.g. grounded theory) to identify themes and sub-themes.   

However, the nature of the relationship between these various themes and sub-themes is often not specified.  

*

==== Causal networks ====
The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers sometime present models with boxes and arrows representing correlations and/or causations&lt;ref&gt;{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}&lt;/ref&gt;. In qualitative research, building grounded theory models is about &quot;[accounting] for not only all the major emergent concepts, themes, and dimensions, but also for their dynamic interrelationships. Speaking in classic boxes-and-arrows terms, this process amounts to assembling the constellation of boxes with a special focus on the arrows.&quot;&lt;ref&gt;{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}&lt;/ref&gt; Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can &quot;feedback&quot; (two variables can influence each other)&lt;ref&gt;{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}&lt;/ref&gt;.

Wikidata includes several properties to describe causal relationships:
* {{Wikidata entity link|P828}}
* {{Wikidata entity link|P1542}}
* {{Wikidata entity link|P1537}}
* {{Wikidata entity link|P1479}} : it is difficult to identify single causes for social phenomenons, many factors having an effect on the subject item will likely be contributing factors

==== Modelling concepts ====
To model concepts related to just transition. We read the selected papers and used them as source to build a knowledge graph in wikidata. For example, the paper {{Wikidata entity link|Q137901182}} mention &quot;Energy democracy is both an ideal and a process&quot;&lt;ref&gt;{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research &amp; Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}&lt;/ref&gt;, we thus entered the wikidata statement {{Wikidata entity link|Q14944319}} is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q840396}}, using the paper as source.

Challenges :
*{{Wikidata entity link|P31}}: concepts have a dual nature because they designate at the same time an idea and the entity that this idea represent.
* {{Wikidata entity link|P3712}}: concepts do not have goals in themselves, but the reality they represent can have goals
* wikidata current items are not really suited to model &quot;meta-research&quot; statements. For example, modelling the idea tha the literature on energy democracy is fragmented would require creating an item representing the energy democracy literature, not just energy democracy in general. Similarly, it can be difficult to model the chronological evolution of the definition of an idea (although it could be technically possible).
* To distinguish the causes of the concepts/discourses and the causes of the phenomenon itself, we used {{Wikidata entity link|P3938}} to indicate the origins of the concept or the movements promoting it.
* To distinguish goals from the process to reach it, we used {{Wikidata entity link|P3712}} to describe ideals and {{Wikidata entity link|P2670}} to describe processes.
* Wikidata does not seem to be the best tool to model quantitative statements, for example, the paper {{Wikidata entity link|Q137901196}} states that &quot;9.8% of the final energy consumed in developing countries comes from modern renewable energy sources&quot;. Including energy data in Wikidata require using or creating specific properties (e.g. {{Wikidata entity link|P6826}})
*

=== Writing ===
To cite articles we used the [[Template:Cite Q|Cite Q template.]] Each reference is an item in Wikidata and the template retrieve the necessary data to generate the citation references below.

== Data == 
{| class=&quot;wikitable sortable&quot;
! QID !! Year !! DOI !! Title
|-
| [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review
|-
| [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review
|-
| [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review
|-
| [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter?
|-
| [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset.
|-
| [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies?
|-
| [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection
|-
| [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development
|-
| [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research
|-
| [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition
|-
| [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning
|-
| [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review
|-
| [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view
|-
| [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory
|-
| [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries
|-
| [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review
|-
| [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions
|-
| [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies
|-
| [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes
|-
| [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation
|-
| [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives
|-
| [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies
|-
| [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda
|-
| [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice?
|-
| [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review
|-
| [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research
|-
| [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape
|-
| [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models
|-
| [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review
|-
| [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions
|-
| [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions
|-
| [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation
|-
| [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings
|-
| [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda
|-
| [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review
|-
| [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework
|-
| [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende
|-
| [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa
|-
| [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities
|-
| [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion
|-
| [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review
|-
| [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights
|-
| [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review
|-
| [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations
|-
| [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance
|-
| [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions
|-
| [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review
|-
| [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice
|-
| [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice
|-
| [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review
|-
| [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review
|-
| [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions
|-
| [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition
|-
| [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy
|-
| [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends
|-
| [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience
|}


== References ==
{{References}}</text>
      <sha1>j9jb9thhhjgkmay995ruthikjm5spfo</sha1>
    </revision>
  </page>
  <page>
    <title>User:Dc.samizdat/Golden chords of the 120-cell</title>
    <ns>2</ns>
    <id>326765</id>
    <revision>
      <id>2807835</id>
      <parentid>2807584</parentid>
      <timestamp>2026-05-06T19:01:52Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2807835</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45123" sha1="grvdc1lhfsnt56jx6t137ygzh5iyw3r" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

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

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

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

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2(2+\sqrt{2})}} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell. In the unit-radius 120-cell, chord &lt;math&gt;c_{15} = \sqrt{2}&lt;/math&gt; occurs in 675 distinct 16-cells. 

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its [[W:Geodesic|geodesic]] orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

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

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing.

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

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}}

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

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices are disjoint circular helixes, and those 16 circular helixes are Clifford parallel objects.

== The 24-cell ==
In 2-space we have the radially equilateral 6-point hexagon, in 3-space we have the radially equilateral 12-point cuboctahedron with 4 hexagonal central planes, and in 4-space we have the radially equilateral 24-point 24-cell with 4 cuboctahedral central hyperplanes and 16 hexagonal central planes.


...

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

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

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>grvdc1lhfsnt56jx6t137ygzh5iyw3r</sha1>
    </revision>
    <revision>
      <id>2807836</id>
      <parentid>2807835</parentid>
      <timestamp>2026-05-06T19:02:57Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2807836</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45122" sha1="0ksfxf7jl9t5l8r618tw3u202obkr6r" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

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

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

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

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2(2+\sqrt{2})} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=\sqrt{2(4 + \sqrt{8})} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell. In the unit-radius 120-cell, chord &lt;math&gt;c_{15} = \sqrt{2}&lt;/math&gt; occurs in 675 distinct 16-cells. 

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its [[W:Geodesic|geodesic]] orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

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

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing.

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

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}}

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

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices are disjoint circular helixes, and those 16 circular helixes are Clifford parallel objects.

== The 24-cell ==
In 2-space we have the radially equilateral 6-point hexagon, in 3-space we have the radially equilateral 12-point cuboctahedron with 4 hexagonal central planes, and in 4-space we have the radially equilateral 24-point 24-cell with 4 cuboctahedral central hyperplanes and 16 hexagonal central planes.


...

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

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

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>0ksfxf7jl9t5l8r618tw3u202obkr6r</sha1>
    </revision>
    <revision>
      <id>2807837</id>
      <parentid>2807836</parentid>
      <timestamp>2026-05-06T19:04:45Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2807837</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45118" sha1="4r80n8n212946yfgb2nomhjxyj71xpw" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

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

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

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

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2(2+\sqrt{2})} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=2 Sqrt[2+\sqrt{2}] \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell. In the unit-radius 120-cell, chord &lt;math&gt;c_{15} = \sqrt{2}&lt;/math&gt; occurs in 675 distinct 16-cells. 

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its [[W:Geodesic|geodesic]] orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

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

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing.

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

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}}

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

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices are disjoint circular helixes, and those 16 circular helixes are Clifford parallel objects.

== The 24-cell ==
In 2-space we have the radially equilateral 6-point hexagon, in 3-space we have the radially equilateral 12-point cuboctahedron with 4 hexagonal central planes, and in 4-space we have the radially equilateral 24-point 24-cell with 4 cuboctahedral central hyperplanes and 16 hexagonal central planes.


...

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

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

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>4r80n8n212946yfgb2nomhjxyj71xpw</sha1>
    </revision>
    <revision>
      <id>2807838</id>
      <parentid>2807837</parentid>
      <timestamp>2026-05-06T19:05:23Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2807838</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45119" sha1="ay8sb93s31vk52qgqr7tgck2suppd2q" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

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

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

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

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2(2+\sqrt{2})} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=2 \sqrt{2+\sqrt{2}} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell. In the unit-radius 120-cell, chord &lt;math&gt;c_{15} = \sqrt{2}&lt;/math&gt; occurs in 675 distinct 16-cells. 

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its [[W:Geodesic|geodesic]] orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

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

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing.

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

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}}

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

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices are disjoint circular helixes, and those 16 circular helixes are Clifford parallel objects.

== The 24-cell ==
In 2-space we have the radially equilateral 6-point hexagon, in 3-space we have the radially equilateral 12-point cuboctahedron with 4 hexagonal central planes, and in 4-space we have the radially equilateral 24-point 24-cell with 4 cuboctahedral central hyperplanes and 16 hexagonal central planes.


...

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

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

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>ay8sb93s31vk52qgqr7tgck2suppd2q</sha1>
    </revision>
    <revision>
      <id>2807839</id>
      <parentid>2807838</parentid>
      <timestamp>2026-05-06T19:12:34Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2807839</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45142" sha1="o3xzr8b4tp63anoun87uabnz9s3icmz" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

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

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

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

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2(2+\sqrt{2})} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=2 \sqrt{2+\sqrt{2}} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell. 

In the unit-radius 120-cell, the great square edge chord &lt;math&gt;c_{15} = \sqrt{2}&lt;/math&gt; occurs in 675 distinct 16-cells.

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its [[W:Geodesic|geodesic]] orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

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

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing.

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

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}}

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

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices are disjoint circular helixes, and those 16 circular helixes are Clifford parallel objects.

== The 24-cell ==
In 2-space we have the radially equilateral 6-point hexagon, in 3-space we have the radially equilateral 12-point cuboctahedron with 4 hexagonal central planes, and in 4-space we have the radially equilateral 24-point 24-cell with 4 cuboctahedral central hyperplanes and 16 hexagonal central planes.


...

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

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

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>o3xzr8b4tp63anoun87uabnz9s3icmz</sha1>
    </revision>
    <revision>
      <id>2807840</id>
      <parentid>2807839</parentid>
      <timestamp>2026-05-06T19:13:53Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2807840</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45156" sha1="dyuxwxqsif5apdcbm2ow3z1nlzvfj7o" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

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

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

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

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2(2+\sqrt{2})} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=2 \sqrt{2+\sqrt{2}} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell or the 120-cell. In the unit-radius 120-cell, the great square edge chord &lt;math&gt;c_{15} = \sqrt{2}&lt;/math&gt; occurs in 675 distinct 16-cells.

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its [[W:Geodesic|geodesic]] orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

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

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing.

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

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}}

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

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices are disjoint circular helixes, and those 16 circular helixes are Clifford parallel objects.

== The 24-cell ==
In 2-space we have the radially equilateral 6-point hexagon, in 3-space we have the radially equilateral 12-point cuboctahedron with 4 hexagonal central planes, and in 4-space we have the radially equilateral 24-point 24-cell with 4 cuboctahedral central hyperplanes and 16 hexagonal central planes.


...

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

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

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>dyuxwxqsif5apdcbm2ow3z1nlzvfj7o</sha1>
    </revision>
    <revision>
      <id>2807841</id>
      <parentid>2807840</parentid>
      <timestamp>2026-05-06T19:16:49Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2807841</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45158" sha1="jvfh3u2ilwpg9kd05sfwg024kgbniz5" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

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

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

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

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2(2+\sqrt{2})} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=2 \sqrt{2+\sqrt{2}} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell or the 120-cell. In the unit-radius 120-cell, the central square edge chord &lt;math&gt;c_{15} = \sqrt{2}&lt;/math&gt; occurs in 675 distinct 16-cells.

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its [[W:Geodesic|geodesic]] orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

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

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing.

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

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}}

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

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices are disjoint circular helixes, and those 16 circular helixes are Clifford parallel objects.

== The 24-cell ==
In 2-space we have the radially equilateral 6-point hexagon, in 3-space we have the radially equilateral 12-point cuboctahedron with 4 hexagonal central planes, and in 4-space we have the radially equilateral 24-point 24-cell with 4 cuboctahedral central hyperplanes and 16 hexagonal central planes.


...

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

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

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>jvfh3u2ilwpg9kd05sfwg024kgbniz5</sha1>
    </revision>
    <revision>
      <id>2807842</id>
      <parentid>2807841</parentid>
      <timestamp>2026-05-06T19:18:23Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2807842</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45172" sha1="cyxmm4gev0jn7nasa2pntr31c3oa1g4" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

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

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

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

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2(2+\sqrt{2})} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=2 \sqrt{2+\sqrt{2}} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell or the 120-cell. In the unit-radius 120-cell, the central square edge chord &lt;math&gt;c_{15} = \sqrt{2}&lt;/math&gt; occurs in 675 distinct (75 disjoint) 16-cells.

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its [[W:Geodesic|geodesic]] orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

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

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing.

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

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}}

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

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices are disjoint circular helixes, and those 16 circular helixes are Clifford parallel objects.

== The 24-cell ==
In 2-space we have the radially equilateral 6-point hexagon, in 3-space we have the radially equilateral 12-point cuboctahedron with 4 hexagonal central planes, and in 4-space we have the radially equilateral 24-point 24-cell with 4 cuboctahedral central hyperplanes and 16 hexagonal central planes.


...

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

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

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>cyxmm4gev0jn7nasa2pntr31c3oa1g4</sha1>
    </revision>
    <revision>
      <id>2807843</id>
      <parentid>2807842</parentid>
      <timestamp>2026-05-06T19:26:16Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2807843</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45080" sha1="ipx28iy5zq4h105jd2hnl0n9fk2udc0" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

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

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

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

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2(2+\sqrt{2})} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=2 \sqrt{2+\sqrt{2}} \approx 3.69552&lt;/math&gt;

In the unit-radius 120-cell, the central square edge chord &lt;math&gt;c_{15} = \sqrt{2}&lt;/math&gt; occurs in 675 distinct (75 disjoint) 16-cells.

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its [[W:Geodesic|geodesic]] orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

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

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing.

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

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}}

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

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices are disjoint circular helixes, and those 16 circular helixes are Clifford parallel objects.

== The 24-cell ==
In 2-space we have the radially equilateral 6-point hexagon, in 3-space we have the radially equilateral 12-point cuboctahedron with 4 hexagonal central planes, and in 4-space we have the radially equilateral 24-point 24-cell with 4 cuboctahedral central hyperplanes and 16 hexagonal central planes.


...

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

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

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>ipx28iy5zq4h105jd2hnl0n9fk2udc0</sha1>
    </revision>
    <revision>
      <id>2807844</id>
      <parentid>2807843</parentid>
      <timestamp>2026-05-06T19:40:59Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2807844</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45168" sha1="9ss9j6sc4et1pluzc0lhvjfxlkrfal0" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

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

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

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

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2(2+\sqrt{2})} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=2 \sqrt{2+\sqrt{2}} \approx 3.69552&lt;/math&gt;

none of which chords except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt; occur in the 16-cell or 120-cell. In the unit-radius 120-cell, the central square edge chord &lt;math&gt;c_{15} = \sqrt{2}&lt;/math&gt; occurs in 675 distinct (75 disjoint) 16-cells.

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its [[W:Geodesic|geodesic]] orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

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

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing.

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

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}}

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

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices are disjoint circular helixes, and those 16 circular helixes are Clifford parallel objects.

== The 24-cell ==
In 2-space we have the radially equilateral 6-point hexagon, in 3-space we have the radially equilateral 12-point cuboctahedron with 4 hexagonal central planes, and in 4-space we have the radially equilateral 24-point 24-cell with 4 cuboctahedral central hyperplanes and 16 hexagonal central planes.


...

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

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

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>9ss9j6sc4et1pluzc0lhvjfxlkrfal0</sha1>
    </revision>
    <revision>
      <id>2807847</id>
      <parentid>2807844</parentid>
      <timestamp>2026-05-06T20:32:22Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2807847</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45166" sha1="0cm42ohffm7pl7e5cv0ui9ykt4x1kwp" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

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

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

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

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2(2+\sqrt{2})} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=2 \sqrt{2+\sqrt{2}} \approx 3.69552&lt;/math&gt;

none of which chords occur in a 16-cell or 120-cell except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt;. In the unit-radius 120-cell, the central square edge chord &lt;math&gt;c_{15} = \sqrt{2}&lt;/math&gt; occurs in 675 distinct (75 disjoint) 16-cells.

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

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its [[W:Geodesic|geodesic]] orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

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

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing.

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

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}}

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

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices are disjoint circular helixes, and those 16 circular helixes are Clifford parallel objects.

== The 24-cell ==
In 2-space we have the radially equilateral 6-point hexagon, in 3-space we have the radially equilateral 12-point cuboctahedron with 4 hexagonal central planes, and in 4-space we have the radially equilateral 24-point 24-cell with 4 cuboctahedral central hyperplanes and 16 hexagonal central planes.


...

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

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

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>0cm42ohffm7pl7e5cv0ui9ykt4x1kwp</sha1>
    </revision>
    <revision>
      <id>2807848</id>
      <parentid>2807847</parentid>
      <timestamp>2026-05-06T20:59:59Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 8-point regular polytopes */</comment>
      <origin>2807848</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45165" sha1="gt5h0m30buhidj7om6vd94g0fws228q" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

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

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

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

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2(2+\sqrt{2})} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=2 \sqrt{2+\sqrt{2}} \approx 3.69552&lt;/math&gt;

none of which chords occur in a 16-cell or 120-cell except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt;. 

In the unit-radius 120-cell, the great square edge chord &lt;math&gt;c_{15} = \sqrt{2}&lt;/math&gt; occurs in 675 distinct (75 disjoint) 16-cells. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its [[W:Geodesic|geodesic]] orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

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

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing.

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

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}}

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

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices are disjoint circular helixes, and those 16 circular helixes are Clifford parallel objects.

== The 24-cell ==
In 2-space we have the radially equilateral 6-point hexagon, in 3-space we have the radially equilateral 12-point cuboctahedron with 4 hexagonal central planes, and in 4-space we have the radially equilateral 24-point 24-cell with 4 cuboctahedral central hyperplanes and 16 hexagonal central planes.


...

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

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

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>gt5h0m30buhidj7om6vd94g0fws228q</sha1>
    </revision>
    <revision>
      <id>2807849</id>
      <parentid>2807848</parentid>
      <timestamp>2026-05-06T21:05:59Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* The 24-cell */</comment>
      <origin>2807849</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="45163" sha1="9495f77qrkitcv49nuiddegmsn2j4tw" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|Draft in progress}}
{{align|center|January 2026 - April 2026}}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

...

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

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

:&lt;math&gt;r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313&lt;/math&gt;

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

:&lt;math&gt;r_3-r_1-r_1=1/r_3 \approx 0.41421&lt;/math&gt;

Notice that &lt;math&gt;1/r_3=\sqrt{2}-1=r_3-2&lt;/math&gt;.

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

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

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

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

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

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

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

The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-orthoplex, the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular 4-polytopes, including the 120-cell, are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell.

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

:&lt;math&gt;r_1=\sqrt{2},r_2=\sqrt{2(2+\sqrt{2})} \approx 2.61313,r_3=2+\sqrt{2} \approx 3.41421,r_4=2 \sqrt{2+\sqrt{2}} \approx 3.69552&lt;/math&gt;

none of which chords occur in a 16-cell or 120-cell except &lt;math&gt;r_1=\sqrt{2}&lt;/math&gt;. 

In the unit-radius 120-cell, the great square edge chord &lt;math&gt;c_{15} = \sqrt{2}&lt;/math&gt; occurs in 675 distinct (75 disjoint) 16-cells. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}}

[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. The two completely orthogonal planes are called invariant rotation planes, because all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements.

The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic isoclinic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every square central plane to its completely orthogonal square central plane, and takes every vertex to its antipodal vertex 180° degrees away. All the vertices move at once, displaced 180° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. The trajectory of each vertex is a one-eighth segment of its [[W:Geodesic|geodesic]] orbit. Its entire orbit traces a circular helix in 4-space, and also traces a great circle in one of the two completely orthogonal invariant rotation planes, as they tilt sideways into each other's plane. When the isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 180° to its antipodal position, but from the new orientation where the vertex is on the opposite side of the 16-cell departing in the opposite direction. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once, and returns to its original position.

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

The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube.

The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular octagon, but the tesseract contains 2 disjoint instances and 4 distinct instances of the skew octagon. We can construct the tesseract the way we constructed the 16-cell, by skewing a planar octagon's edges so they become edges of the 4-polytope. Because the tesseract has 16 vertices we will need two planar octagons, and to start we must embedded them in 4-space as completely orthogonal planes that intersect at only one point, their common center. Because the tesseract is radially equilateral (unlike the 16-cell), to build a unit-radius tesseract we start with our original octagon of unit-edge length, rather than the octagon of edge length &lt;small&gt;&lt;math&gt;\sqrt{2}&lt;/math&gt;&lt;/small&gt; that we needed to build the unit-radius 16-cell.

For our tesseract construction we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes. Provided the planes were completely orthogonal in 4-space and we skewed them both the same way, the 16 vertices will be the vertices of a tesseract with half of its 32 edges missing.

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

Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common and their corresponding vertices 180° apart. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation with their corresponding vertices 90° apart, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.}}

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

We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices are disjoint circular helixes, and those 16 circular helixes are Clifford parallel objects.

== The 24-cell ==
In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 4 cuboctahedral central hyperplanes and 16 hexagonal central planes.


...

== The 600-cell ==

...

== Finally the 120-cell ==

...

== Conclusions ==

Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional regular polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords.

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

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

== Notes ==
{{Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }}
* {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson &amp; Sazdanovich|2004}} }}
* {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff &amp; Adamson|2004}} }}
* {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }}
{{Refend}}</text>
      <sha1>9495f77qrkitcv49nuiddegmsn2j4tw</sha1>
    </revision>
  </page>
  <page>
    <title>User:Atcovi/OGM &amp; Suicide/The Paper</title>
    <ns>2</ns>
    <id>329353</id>
    <revision>
      <id>2807833</id>
      <parentid>2807695</parentid>
      <timestamp>2026-05-06T17:55:20Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>/* Mechanisms */</comment>
      <origin>2807833</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8354" sha1="8crh49u3cuqd0vrg9grhyq07s31hqhc" xml:space="preserve">''Better translated as the &quot;thinking space&quot; vs. an actual paper.''

''What we doing?'' Integrating OGM → suicidal ideation within a structured model (IMV + mechanisms).

==Introduction==
'''[[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory]]''' (OGM) describes a reduced ability to recall specific events in one's autobiographical memory. For example, one may remember attending a birthday party at some point in their life, but they could not uniquely recall a specific instance of attending a birthday party. OGM has been empirically associated with depression, with depressed individuals reporting higher levels of OGM than non-depressed individuals&lt;ref&gt;{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}&lt;/ref&gt;. Given the association of depression and suicidal ideation&lt;ref&gt;{{Cite journal|last=Chachamovich|first=Eduardo|last2=Stefanello|first2=Sabrina|last3=Botega|first3=Neury|last4=Turecki|first4=Gustavo|date=2009-05|title=[Which are the recent clinical findings regarding the association between depression and suicide?]|url=https://pubmed.ncbi.nlm.nih.gov/19565147|journal=Revista Brasileira De Psiquiatria (Sao Paulo, Brazil: 1999)|volume=31 Suppl 1|pages=S18–25|doi=10.1590/s1516-44462009000500004|issn=1516-4446|pmid=19565147}}&lt;/ref&gt;, utilizing suicide models, such as the '''Integrated Motivational-Volitional (IMV) model''', provides a theoretical cognitive framework to increase understanding of where OGM may fit in the escalation to suicidal ideation and/or suicide.

The IMV model portrays suicidal behavior as an escalating, behavioral process divided into three phases: pre-motivational phase, motivational phase, and volitional phase. The motivational phase is characterized by suicidal ideation formation, where feelings of entrapment (described as a &quot;proximal [predictor] of suicidal ideation&quot;&lt;ref&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational-volitional model of suicidal behaviour|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/|journal=Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=1471-2970|pmc=6053985|pmid=30012735}}&lt;/ref&gt;), poor problem-solving abilities, brooding, and interpersonal vulnerabilities (thwarted belongingness and perceived burdensomeness) may transition the individual to the volitional phase. When looking at the IMV model and assessing where OGM could play a part in the transition to suicidal ideation, OGM may impair problem-solving capabilities and the ability to learn from the past through reduced retrieval of specific past experiences, leading to hopelessness&lt;ref&gt;{{Cite journal|last=Jiang|first=Wen|last2=Hu|first2=Guangtao|last3=Zhang|first3=Jingxuan|last4=Chen|first4=Ken|last5=Fan|first5=Dongni|last6=Feng|first6=Zhengzhi|date=2020-10-12|title=Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people|url=https://doi.org/10.1186/s12888-020-02877-6|journal=BMC Psychiatry|language=en|volume=20|issue=1|pages=501|doi=10.1186/s12888-020-02877-6|issn=1471-244X|pmc=7549224|pmid=33046032}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5}}&lt;/ref&gt;.

The current literature aims to shed light on a neglected niche of suicide research: autobiographical memory. Despite the overwhelming research suggesting correlations between OGM and depression and suicidal ideation, research has not thoroughly explored OGM's exact role in a cognitive, theoretical framework of suicidal ideation (specifically within the IMV model). By conducting a narrative review and integrating research on OGM's role in suicidal ideation, this paper furthers understanding on OGM's role in suicidal ideation. 

==OGM as Vulnerability==
Evidence suggests that OGM is a cognitive vulnerability associated with depression and suicidal ideation, though its predictive relevance may vary depending on the population. 

A meta analysis performed by Sumner et. al (2010) found that OGM accounted for about 1-2% of the variance in depressive symptoms at follow-up&lt;ref&gt;{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}&lt;/ref&gt;. A 2020 study found that OGM was associated with depressed patients' current suicidal ideation state and worse-point suicidal ideation, while OGM affected the healthy patients' worse-point suicidal ideation&lt;ref&gt;{{Cite journal|last=Jiang|first=Wen|last2=Hu|first2=Guangtao|last3=Zhang|first3=Jingxuan|last4=Chen|first4=Ken|last5=Fan|first5=Dongni|last6=Feng|first6=Zhengzhi|date=2020-10-12|title=Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people|url=https://doi.org/10.1186/s12888-020-02877-6|journal=BMC Psychiatry|language=en|volume=20|issue=1|pages=501|doi=10.1186/s12888-020-02877-6|issn=1471-244X|pmc=7549224|pmid=33046032}}&lt;/ref&gt;. However, Crane et. al (2016) conducted a longitudinal study of n≈5800 adolescents from ages 13 to 16 and found that OGM was not significantly associated with depression and did not moderate the effect of life events, suggesting OGM may not be as generalizable to community samples vs. high-risk populations&lt;ref&gt;{{Cite journal|last=Crane|first=Catherine|last2=Heron|first2=Jon|last3=Gunnell|first3=David|last4=Lewis|first4=Glyn|last5=Evans|first5=Jonathan|last6=Williams|first6=J. Mark G.|date=2016|title=Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort study|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/|journal=Memory (Hove, England)|volume=24|issue=3|pages=348–363|doi=10.1080/09658211.2015.1008014|issn=1464-0686|pmc=4743605|pmid=25716137}}&lt;/ref&gt;. Even though the findings indicated that OGM was not significantly associated with depression, the study highlights that OGM may function as a vulnerability within high-risk populations. This suggests that OGM's predictive relevance may depend on the risk-level of the population.

OGM persists even past depression, as found in Hallford et. al (2022). A meta-analysis indicated that participants with remitted depression continue to experience small to moderate deficiencies in being able to recall &quot;specific, event-level personal memories&quot;. This indicates that OGM isn't merely a symptom of depression, but may function as a risk factor for future depressive episodes&lt;ref&gt;{{Cite journal|last=Hallford|first=David John|last2=Rusanov|first2=Danielle|last3=Yeow|first3=Joseph J. E.|last4=Barry|first4=Tom Joseph|date=2022-09|title=Reduced specificity and increased overgenerality of autobiographical memory persist as cognitive vulnerabilities in remitted major depression: A meta-analysis|url=https://pubmed.ncbi.nlm.nih.gov/36129959|journal=Clinical Psychology &amp; Psychotherapy|volume=29|issue=5|pages=1515–1529|doi=10.1002/cpp.2786|issn=1099-0879|pmc=9828164|pmid=36129959}}&lt;/ref&gt;.    

Considering the above, the analysis suggests that OGM may function as a cognitive vulnerability that is further exacerbated in high-risk populations vs. the general population.    

==Mechanisms==
'''Structure'''

# Specific memories help problem solving, OGM impairs this
# OGM makes ppl brood/traps 'em in brooding/rummination 
# Tie to IMV model + suicidal ideation formation

==OGM → Suicidal ideation (CORE)==
==Contradictions / Nuances==
==Conclusion==

== References ==
{{Reflist}}
[[Category:Atcovi/OGM &amp; Suicide Poster]]</text>
      <sha1>8crh49u3cuqd0vrg9grhyq07s31hqhc</sha1>
    </revision>
    <revision>
      <id>2807834</id>
      <parentid>2807833</parentid>
      <timestamp>2026-05-06T18:02:18Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>/* Mechanisms */</comment>
      <origin>2807834</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="8356" sha1="e6ebwaiinoum3mj1ggxzlg2h5vip7ld" xml:space="preserve">''Better translated as the &quot;thinking space&quot; vs. an actual paper.''

''What we doing?'' Integrating OGM → suicidal ideation within a structured model (IMV + mechanisms).

==Introduction==
'''[[w:Overgeneral_autobiographical_memory|Overgeneral autobiographical memory]]''' (OGM) describes a reduced ability to recall specific events in one's autobiographical memory. For example, one may remember attending a birthday party at some point in their life, but they could not uniquely recall a specific instance of attending a birthday party. OGM has been empirically associated with depression, with depressed individuals reporting higher levels of OGM than non-depressed individuals&lt;ref&gt;{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}&lt;/ref&gt;. Given the association of depression and suicidal ideation&lt;ref&gt;{{Cite journal|last=Chachamovich|first=Eduardo|last2=Stefanello|first2=Sabrina|last3=Botega|first3=Neury|last4=Turecki|first4=Gustavo|date=2009-05|title=[Which are the recent clinical findings regarding the association between depression and suicide?]|url=https://pubmed.ncbi.nlm.nih.gov/19565147|journal=Revista Brasileira De Psiquiatria (Sao Paulo, Brazil: 1999)|volume=31 Suppl 1|pages=S18–25|doi=10.1590/s1516-44462009000500004|issn=1516-4446|pmid=19565147}}&lt;/ref&gt;, utilizing suicide models, such as the '''Integrated Motivational-Volitional (IMV) model''', provides a theoretical cognitive framework to increase understanding of where OGM may fit in the escalation to suicidal ideation and/or suicide.

The IMV model portrays suicidal behavior as an escalating, behavioral process divided into three phases: pre-motivational phase, motivational phase, and volitional phase. The motivational phase is characterized by suicidal ideation formation, where feelings of entrapment (described as a &quot;proximal [predictor] of suicidal ideation&quot;&lt;ref&gt;{{Cite journal|last=O'Connor|first=Rory C.|last2=Kirtley|first2=Olivia J.|date=2018-09-05|title=The integrated motivational-volitional model of suicidal behaviour|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC6053985/|journal=Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences|volume=373|issue=1754|pages=20170268|doi=10.1098/rstb.2017.0268|issn=1471-2970|pmc=6053985|pmid=30012735}}&lt;/ref&gt;), poor problem-solving abilities, brooding, and interpersonal vulnerabilities (thwarted belongingness and perceived burdensomeness) may transition the individual to the volitional phase. When looking at the IMV model and assessing where OGM could play a part in the transition to suicidal ideation, OGM may impair problem-solving capabilities and the ability to learn from the past through reduced retrieval of specific past experiences, leading to hopelessness&lt;ref&gt;{{Cite journal|last=Jiang|first=Wen|last2=Hu|first2=Guangtao|last3=Zhang|first3=Jingxuan|last4=Chen|first4=Ken|last5=Fan|first5=Dongni|last6=Feng|first6=Zhengzhi|date=2020-10-12|title=Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people|url=https://doi.org/10.1186/s12888-020-02877-6|journal=BMC Psychiatry|language=en|volume=20|issue=1|pages=501|doi=10.1186/s12888-020-02877-6|issn=1471-244X|pmc=7549224|pmid=33046032}}&lt;/ref&gt;&lt;ref&gt;{{Cite book|url=https://doi.org/10.1017/9781316563205|title=The Neuroscience of Suicidal Behavior|last=van Heeringen|first=Kees|date=2018-08-23|publisher=Cambridge University Press|isbn=978-1-316-56320-5}}&lt;/ref&gt;.

The current literature aims to shed light on a neglected niche of suicide research: autobiographical memory. Despite the overwhelming research suggesting correlations between OGM and depression and suicidal ideation, research has not thoroughly explored OGM's exact role in a cognitive, theoretical framework of suicidal ideation (specifically within the IMV model). By conducting a narrative review and integrating research on OGM's role in suicidal ideation, this paper furthers understanding on OGM's role in suicidal ideation. 

==OGM as Vulnerability==
Evidence suggests that OGM is a cognitive vulnerability associated with depression and suicidal ideation, though its predictive relevance may vary depending on the population. 

A meta analysis performed by Sumner et. al (2010) found that OGM accounted for about 1-2% of the variance in depressive symptoms at follow-up&lt;ref&gt;{{Cite journal|last=Sumner|first=Jennifer A.|last2=Griffith|first2=James W.|last3=Mineka|first3=Susan|date=2010-07|title=Overgeneral autobiographical memory as a predictor of the course of depression: a meta-analysis|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC2878838/|journal=Behaviour Research and Therapy|volume=48|issue=7|pages=614–625|doi=10.1016/j.brat.2010.03.013|issn=1873-622X|pmc=2878838|pmid=20399418}}&lt;/ref&gt;. A 2020 study found that OGM was associated with depressed patients' current suicidal ideation state and worse-point suicidal ideation, while OGM affected the healthy patients' worse-point suicidal ideation&lt;ref&gt;{{Cite journal|last=Jiang|first=Wen|last2=Hu|first2=Guangtao|last3=Zhang|first3=Jingxuan|last4=Chen|first4=Ken|last5=Fan|first5=Dongni|last6=Feng|first6=Zhengzhi|date=2020-10-12|title=Distinct effects of over-general autobiographical memory on suicidal ideation among depressed and healthy people|url=https://doi.org/10.1186/s12888-020-02877-6|journal=BMC Psychiatry|language=en|volume=20|issue=1|pages=501|doi=10.1186/s12888-020-02877-6|issn=1471-244X|pmc=7549224|pmid=33046032}}&lt;/ref&gt;. However, Crane et. al (2016) conducted a longitudinal study of n≈5800 adolescents from ages 13 to 16 and found that OGM was not significantly associated with depression and did not moderate the effect of life events, suggesting OGM may not be as generalizable to community samples vs. high-risk populations&lt;ref&gt;{{Cite journal|last=Crane|first=Catherine|last2=Heron|first2=Jon|last3=Gunnell|first3=David|last4=Lewis|first4=Glyn|last5=Evans|first5=Jonathan|last6=Williams|first6=J. Mark G.|date=2016|title=Adolescent over-general memory, life events and mental health outcomes: Findings from a UK cohort study|url=https://pmc.ncbi.nlm.nih.gov/articles/PMC4743605/|journal=Memory (Hove, England)|volume=24|issue=3|pages=348–363|doi=10.1080/09658211.2015.1008014|issn=1464-0686|pmc=4743605|pmid=25716137}}&lt;/ref&gt;. Even though the findings indicated that OGM was not significantly associated with depression, the study highlights that OGM may function as a vulnerability within high-risk populations. This suggests that OGM's predictive relevance may depend on the risk-level of the population.

OGM persists even past depression, as found in Hallford et. al (2022). A meta-analysis indicated that participants with remitted depression continue to experience small to moderate deficiencies in being able to recall &quot;specific, event-level personal memories&quot;. This indicates that OGM isn't merely a symptom of depression, but may function as a risk factor for future depressive episodes&lt;ref&gt;{{Cite journal|last=Hallford|first=David John|last2=Rusanov|first2=Danielle|last3=Yeow|first3=Joseph J. E.|last4=Barry|first4=Tom Joseph|date=2022-09|title=Reduced specificity and increased overgenerality of autobiographical memory persist as cognitive vulnerabilities in remitted major depression: A meta-analysis|url=https://pubmed.ncbi.nlm.nih.gov/36129959|journal=Clinical Psychology &amp; Psychotherapy|volume=29|issue=5|pages=1515–1529|doi=10.1002/cpp.2786|issn=1099-0879|pmc=9828164|pmid=36129959}}&lt;/ref&gt;.    

Considering the above, the analysis suggests that OGM may function as a cognitive vulnerability that is further exacerbated in high-risk populations vs. the general population.    

==Mechanisms==
'''Structure'''

# Specific memories help problem solving, OGM impairs this
# OGM makes ppl brood/traps 'em in brooding or rummination
# Tie to IMV model + suicidal ideation formation

==OGM → Suicidal ideation (CORE)==
==Contradictions / Nuances==
==Conclusion==

== References ==
{{Reflist}}
[[Category:Atcovi/OGM &amp; Suicide Poster]]</text>
      <sha1>e6ebwaiinoum3mj1ggxzlg2h5vip7ld</sha1>
    </revision>
  </page>
  <page>
    <title>User:Emilyniven/sandbox</title>
    <ns>2</ns>
    <id>329501</id>
    <revision>
      <id>2807804</id>
      <timestamp>2026-05-06T13:19:16Z</timestamp>
      <contributor>
        <username>Emilyniven</username>
        <id>3070713</id>
      </contributor>
      <comment>made some tweaks</comment>
      <origin>2807804</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="253" sha1="5kvoxhfhy9xsysjvlketfi95ycg5sew" xml:space="preserve">me and dakota are in business rn... she is talking to her boyfriend alfie lol. i am so bored iv finished my course work. anyways if any of my future employees see this it proves i am hard working and committed as i have already completed my course work!</text>
      <sha1>5kvoxhfhy9xsysjvlketfi95ycg5sew</sha1>
    </revision>
  </page>
  <page>
    <title>File:VLSI.Arith.2A.CLA.20260506.pdf</title>
    <ns>6</ns>
    <id>329503</id>
    <revision>
      <id>2807810</id>
      <timestamp>2026-05-06T13:32:45Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=Carry Lookahead Adders 2A traditional (20260506 - 20260505)
|Source={{own|Young1lim}}
|Date=2026-05-06
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2807810</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="282" sha1="c89w6mu8vrdil31kskbraadje74iwf8" xml:space="preserve">== Summary ==
{{Information
|Description=Carry Lookahead Adders 2A traditional (20260506 - 20260505)
|Source={{own|Young1lim}}
|Date=2026-05-06
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
      <sha1>c89w6mu8vrdil31kskbraadje74iwf8</sha1>
    </revision>
  </page>
  <page>
    <title>File:C04.SA0.PtrOperator.1A.20260506.pdf</title>
    <ns>6</ns>
    <id>329504</id>
    <revision>
      <id>2807813</id>
      <timestamp>2026-05-06T13:42:12Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=C04.SA0: Address and Dereference Operators (20260506 - 20260505)
|Source={{own|Young1lim}}
|Date=2026-05-06
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2807813</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="287" sha1="s1auelaplllkmt8g2fhg951kil6mf69" xml:space="preserve">== Summary ==
{{Information
|Description=C04.SA0: Address and Dereference Operators (20260506 - 20260505)
|Source={{own|Young1lim}}
|Date=2026-05-06
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
      <sha1>s1auelaplllkmt8g2fhg951kil6mf69</sha1>
    </revision>
  </page>
  <page>
    <title>File:Laurent.5.Permutation.6C.20260506.pdf</title>
    <ns>6</ns>
    <id>329505</id>
    <revision>
      <id>2807815</id>
      <timestamp>2026-05-06T13:53:21Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=Laurent.5: Permutation 6C (2026506 - 20260505)
|Source={{own|Young1lim}}
|Date=2026-05-06
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2807815</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="269" sha1="f8fvqmz8jmfl1vuqg481pheq1roek07" xml:space="preserve">== Summary ==
{{Information
|Description=Laurent.5: Permutation 6C (2026506 - 20260505)
|Source={{own|Young1lim}}
|Date=2026-05-06
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
      <sha1>f8fvqmz8jmfl1vuqg481pheq1roek07</sha1>
    </revision>
  </page>
  <page>
    <title>File:CP.FileCntl.A.20260506.pdf</title>
    <ns>6</ns>
    <id>329506</id>
    <revision>
      <id>2807821</id>
      <timestamp>2026-05-06T15:51:39Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=Copilot: File Control A. Overview (20260506 - 20260505)
|Source={{own|Young1lim}}
|Date=2026-05-06
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2807821</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="278" sha1="tk2t2bkptoditdez1qsxsp9cbnoq5pt" xml:space="preserve">== Summary ==
{{Information
|Description=Copilot: File Control A. Overview (20260506 - 20260505)
|Source={{own|Young1lim}}
|Date=2026-05-06
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
      <sha1>tk2t2bkptoditdez1qsxsp9cbnoq5pt</sha1>
    </revision>
  </page>
  <page>
    <title>File:Sample.TappedDelay.20260506.pdf</title>
    <ns>6</ns>
    <id>329507</id>
    <revision>
      <id>2807823</id>
      <timestamp>2026-05-06T16:34:49Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=Sample: Tapped Delay (20260506 - 20260505)
|Source={{own|Young1lim}}
|Date=2026-05-06
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2807823</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="265" sha1="q5bvaec2i2xnv60lbt8fvonjpzpd1am" xml:space="preserve">== Summary ==
{{Information
|Description=Sample: Tapped Delay (20260506 - 20260505)
|Source={{own|Young1lim}}
|Date=2026-05-06
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
      <sha1>q5bvaec2i2xnv60lbt8fvonjpzpd1am</sha1>
    </revision>
  </page>
  <page>
    <title>File:DD3.A5.FFTiming.20260504.pdf</title>
    <ns>6</ns>
    <id>329508</id>
    <revision>
      <id>2807825</id>
      <timestamp>2026-05-06T17:37:46Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=FF Timing (20260504 - 20260430)
|Source={{own|Young1lim}}
|Date=2026-05-06
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2807825</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="254" sha1="ji2ok4p5ztad9u1ixbvru84kiv5ivw9" xml:space="preserve">== Summary ==
{{Information
|Description=FF Timing (20260504 - 20260430)
|Source={{own|Young1lim}}
|Date=2026-05-06
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
      <sha1>ji2ok4p5ztad9u1ixbvru84kiv5ivw9</sha1>
    </revision>
  </page>
  <page>
    <title>File:DD3.A5.FFTiming.20260505.pdf</title>
    <ns>6</ns>
    <id>329509</id>
    <revision>
      <id>2807827</id>
      <timestamp>2026-05-06T17:38:53Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=FF Timing (20260505 - 20260504)
|Source={{own|Young1lim}}
|Date=2026-05-06
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2807827</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="254" sha1="qcntobf5dckipk42oopqrlugrqequyi" xml:space="preserve">== Summary ==
{{Information
|Description=FF Timing (20260505 - 20260504)
|Source={{own|Young1lim}}
|Date=2026-05-06
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
      <sha1>qcntobf5dckipk42oopqrlugrqequyi</sha1>
    </revision>
  </page>
  <page>
    <title>File:DD3.A5.FFTiming.20260506.pdf</title>
    <ns>6</ns>
    <id>329510</id>
    <revision>
      <id>2807829</id>
      <timestamp>2026-05-06T17:39:40Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=FF Timing (20260506 - 20260505)
|Source={{own|Young1lim}}
|Date=2026-05-06
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2807829</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="254" sha1="czo73yxs6ob6vdlqx2bngztv09jggha" xml:space="preserve">== Summary ==
{{Information
|Description=FF Timing (20260506 - 20260505)
|Source={{own|Young1lim}}
|Date=2026-05-06
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
      <sha1>czo73yxs6ob6vdlqx2bngztv09jggha</sha1>
    </revision>
  </page>
  <page>
    <title>Virendra Mohan Dar</title>
    <ns>0</ns>
    <id>329511</id>
    <revision>
      <id>2807831</id>
      <timestamp>2026-05-06T17:45:48Z</timestamp>
      <contributor>
        <username>MNTPicker</username>
        <id>3070751</id>
      </contributor>
      <comment>Created new page- bn wiki</comment>
      <origin>2807831</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5373" sha1="2w91xti4g8aycbxyw3qzdbs9v8zuhq2" xml:space="preserve">[[File:Oil Portrait of Maharaja Virendra Mohan Dar of Akhnoor.jpg|thumb|Claude-Sterling Oil Portrait of Maharaja Virendra Mohan Dar of Akhnoor- ''The Maharaja at 26'']]
== Maharaja Virendra Mohan Dar - Founder of the Dar Raj ==
[[W:Maharaja|Maharaja]] Virendra Vasudev Mohan Dar, otherwise known as the effective founder of the Dar Raj's political and ceremonial standing, was born on Thursday, the 14th of September 1758, in the ancestral quarters of the Akhnoor region of Kashmir. He was the eldest son of Ram Hari Mohan Dar and Smt. Annapurna Devi, the third daughter of the esteemed merchant Pandit Narayan Kaul of Srinagar. His lineage traces back to his grandfather, Hari Krishna Mohan Dar (1687–1768), a saffron merchant and learned Kashmiri Pandit who established the family’s zamindari foundations in the late 17th century. 

From his earliest years, Virendra exhibited a discerning mind and a keen disposition for learning. He was educated under several specialized tutors: Pandit Madhusudan Kaul (classical literature, Sanskrit, Persian, and land management), Pandit Gopesh Raina (arithmetic, accounts, and revenue management), and Pandit Jagannath Bhat (Durrani administrative customs and local jurisprudence). By the age of ten, he was already distinguished for his recitals of historical and sacred texts, and by seventeen, he was accompanying his father on tours of the family's vast estates in both Kashmir and Bengal, including Dhamrai and Char Talibari

== Accession and the Title of Maharaja ==

Upon the death of his father in 1778, Virendra Mohan Dar assumed full responsibility for the administration of the Dar Raj estates.His accession occurred during a period of significant political flux as the [[W:Durrani Empire|Durrani Empire]] consolidated power in the Punjab and Kashmir. In 1780, following his judicious resolution of disputes among neighboring zamindars, the court of [[W:Ahmad Shah Durrani|Ahmad Shah Durrani]] conferred upon him the prestigious hereditary title of &quot;Maharaja&quot;.

The formal investiture took place in the autumn of 1780 in the principal hall of the Akhnoor estate.The Maharaja was presented with robes of state and a ceremonial sword, and he pledged to govern with fairness and diligence. To consolidate his authority, he convened a formal Assembly of Zamindars at Akhnur in 1781 to settle boundary disputes and restate revenue obligations. He was known for a &quot;calculated exercise of power,&quot; notably seen in 1782 when he resolved a case of revenue defiance through public inquiry and surveyors rather than armed force

== The Migration to Bengal ==

By the late 18th century, the political stability of the northern territories declined. Provincial governors began prioritizing immediate revenue extraction, and the Akhnoor holdings faced increasing pressure from irregular levies and armed groups associated with local power brokers. In response, the Maharaja implemented a strategic reorientation, gradually shifting the center of his administration to the fertile and more stable plains of [[W:Bengal|Bengal]].

This transition was finalized by a natural calamity in the late 1790s. An exceptionally severe flooding of the [[W:Padma River|Padma River]] resulted in the rapid submergence of the Char Talibari estate, erasing established boundaries and rendering the former seat uninhabitable. Consequently, in 1801, the Maharaja established the Nannar Rajbari (later known as the Dhar Zamindar Bari) in the Dhamrai region. The new residence featured thick brick walls bound with lime-surki mortar and included the Maharaja Virendra Sagar, a large reservoir providing water for both the household and local irrigation.

== Courtly Life and Administration ==

Courtly life at Nannar was governed by a disciplined structure, distinguishing between public functions in the outer courts and private life in the inner quarters. Daily routines included administrative sessions where estate officers presented accounts of cultivation and revenue. The Maharaja was known to dress in fine muslin and silk robes, and the meals served at court reflected a blending of both Kashmiri and Bengali culinary influences

The administration of the Bengal estates—including villages such as ''Rajrajeshwar'', ''Rowail'', ''Sharifbagh'', and ''Ashulia''—was conducted with diligence. The Maharaja personally inspected irrigation works and canals, ensuring that the welfare of the cultivators was protected.

== Later Years and Succession ==

In his later years, the Maharaja withdrew from daily arduous labor but remained steadfast in his supervision of revenue and justice. In 1820, his health began to decline due to a malady of the stomach. Maharaja Virendra Mohan Dar passed away on the 3rd of February, 1821, at the age of sixty-two.

The legacy of the Dar Raj was carried forward by his sons, Raja Mukund Mohan Dar and Bhupendra Mohan Dhar. His lineage continued to produce distinguished figures, including Rai Bahadur Hara Mohan Dhar (a barrister of the Middle Temple), Justice Mohini Mohan Dhar, Judge and former Dewan of Mayurbhanj, Satyendra Mohan Dhar, C.I.E. I.C.S.

== See also ==
Other resources at the [[School:History|School of History]]:
*[[W:Kashmiri Pandits|History of Kashmiri Pandits]]
*[[W:Zamindar|The Zamindari System of Bengal]]
*[[W:Durrani Empire|The Durrani Empire in India]]

[[Category:History of India]]</text>
      <sha1>2w91xti4g8aycbxyw3qzdbs9v8zuhq2</sha1>
    </revision>
  </page>
  <page>
    <title>Talk:Virendra Mohan Dar</title>
    <ns>1</ns>
    <id>329512</id>
    <revision>
      <id>2807832</id>
      <timestamp>2026-05-06T17:47:46Z</timestamp>
      <contributor>
        <username>MNTPicker</username>
        <id>3070751</id>
      </contributor>
      <comment>Created page with &quot;{{talkheader}}&quot;</comment>
      <origin>2807832</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14" sha1="hcd9aq74588nwd90g7oo8u5m36esld6" xml:space="preserve">{{talkheader}}</text>
      <sha1>hcd9aq74588nwd90g7oo8u5m36esld6</sha1>
    </revision>
  </page>
  <page>
    <title>User:HannahTadea/sandbox</title>
    <ns>2</ns>
    <id>329514</id>
    <revision>
      <id>2807853</id>
      <timestamp>2026-05-07T05:36:39Z</timestamp>
      <contributor>
        <username>HannahTadea</username>
        <id>3070835</id>
      </contributor>
      <comment>Created page with &quot;== Business Maintenance Plan ==  === Introduction === This Business Maintenance Plan explains the steps and procedures needed to maintain business operations effectively and efficiently. The goal of this plan is to ensure that the business remains organized, secure, and prepared for possible problems or emergencies.  === Maintenance Steps ===  # Check business equipment regularly. # Update business records and documents. # Monitor inventory and supplies. # Maintain clean...&quot;</comment>
      <origin>2807853</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1048" sha1="9k25pyibghoafve9k17a26pe73yj3lu" xml:space="preserve">== Business Maintenance Plan ==

=== Introduction ===
This Business Maintenance Plan explains the steps and procedures needed to maintain business operations effectively and efficiently. The goal of this plan is to ensure that the business remains organized, secure, and prepared for possible problems or emergencies.

=== Maintenance Steps ===

# Check business equipment regularly.
# Update business records and documents.
# Monitor inventory and supplies.
# Maintain cleanliness and safety in the workplace.
# Back up important digital files and information.
# Review employee responsibilities and schedules.
# Prepare emergency contacts and backup plans.

=== Important Reminders ===

* '''Always maintain accurate records.'''
* ''Keep all important files secured and updated.''
* Follow proper communication within the business.
* Ensure that all systems are functioning properly.

=== Conclusion ===
A proper business maintenance plan helps a business operate smoothly, avoid unnecessary problems, and stay prepared for unexpected situations.</text>
      <sha1>9k25pyibghoafve9k17a26pe73yj3lu</sha1>
    </revision>
    <revision>
      <id>2807854</id>
      <parentid>2807853</parentid>
      <timestamp>2026-05-07T05:38:48Z</timestamp>
      <contributor>
        <username>HannahTadea</username>
        <id>3070835</id>
      </contributor>
      <minor />
      <comment>italicized the important reminders part</comment>
      <origin>2807854</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="1054" sha1="dpie1dvom0zce6mtf6zv187w87f8gx2" xml:space="preserve">== Business Maintenance Plan ==

=== Introduction ===
This Business Maintenance Plan explains the steps and procedures needed to maintain business operations effectively and efficiently. The goal of this plan is to ensure that the business remains organized, secure, and prepared for possible problems or emergencies.

=== Maintenance Steps ===

# Check business equipment regularly.
# Update business records and documents.
# Monitor inventory and supplies.
# Maintain cleanliness and safety in the workplace.
# Back up important digital files and information.
# Review employee responsibilities and schedules.
# Prepare emergency contacts and backup plans.

=== Important Reminders ===

* ''Always maintain accurate records.''
* ''Keep all important files secured and updated.''
* ''Follow proper communication within the business.''
* ''Ensure that all systems are functioning properly.''

=== Conclusion ===
A proper business maintenance plan helps a business operate smoothly, avoid unnecessary problems, and stay prepared for unexpected situations.</text>
      <sha1>dpie1dvom0zce6mtf6zv187w87f8gx2</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Sandbox/header template</title>
    <ns>4</ns>
    <id>329515</id>
    <revision>
      <id>2807855</id>
      <timestamp>2026-05-07T05:45:17Z</timestamp>
      <contributor>
        <username>HannahTadea</username>
        <id>3070835</id>
      </contributor>
      <comment>Created page with &quot;== The Unbreakable Business Plan ==  === Business Name === SwiftCart Online Shop  === Introduction === This Business Maintenance Plan is designed to protect the company’s operations, customer information, and online services from technical problems such as system crashes, hacking, and data loss. The goal of this plan is to ensure that the business can continue operating smoothly even during emergencies.  === Standard Operating Procedure (SOP) ===  ==== 1. Backup Schedu...&quot;</comment>
      <origin>2807855</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2534" sha1="cyz5ym4byvh6sq2y9sxshl8dzpkiptv" xml:space="preserve">== The Unbreakable Business Plan ==

=== Business Name ===
SwiftCart Online Shop

=== Introduction ===
This Business Maintenance Plan is designed to protect the company’s operations, customer information, and online services from technical problems such as system crashes, hacking, and data loss. The goal of this plan is to ensure that the business can continue operating smoothly even during emergencies.

=== Standard Operating Procedure (SOP) ===

==== 1. Backup Schedule ====
* Customer and transaction data must be backed up every hour.
* A full system backup must be performed every day at 12:00 midnight.
* Backup files must be stored in cloud storage and an external hard drive.

'''Purpose:'''
This prevents the loss of important customer and business information during system failures.

==== 2. Security Check ====
* All employees must use strong passwords with letters, numbers, and symbols.
* Two-factor authentication must be enabled for all admin accounts.
* The system must be checked weekly for viruses and suspicious activities.
* Customer information must remain confidential and protected.

'''Purpose:'''
This helps prevent hacking, data theft, and unauthorized access.

==== 3. System Maintenance ====
* The website and application must be checked every morning before operations begin.
* Software updates must be installed immediately after testing.
* Broken links, payment errors, and loading problems must be fixed immediately.

'''Purpose:'''
Regular maintenance keeps the system fast, stable, and reliable.

==== 4. Emergency Response Plan ====
# Inform the IT Team immediately.
# Switch to backup servers within 10 minutes.
# Post announcements on social media to inform customers.
# Recover lost data using backup files.
# Document the problem and solution in the maintenance log.

'''Purpose:'''
This minimizes company losses and restores operations quickly.

==== 5. Emergency Contact List ====
* Operations Manager – 09123456789
* IT Support Team – 09987654321
* Security Officer – 09771234567

=== Audit Trail / Change Log ===
* Added backup schedule for customer data.
* Updated security procedures for admin accounts.
* Improved emergency response steps for faster recovery.

=== Conclusion ===
A proper Business Maintenance Plan helps the company avoid major losses, protect customer data, and maintain smooth business operations. Through regular maintenance, proper documentation, and emergency planning, the business can continue operating successfully even during unexpected situations.</text>
      <sha1>cyz5ym4byvh6sq2y9sxshl8dzpkiptv</sha1>
    </revision>
  </page>
</mediawiki>